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VroRATION TESTING BY DESIGN: 

EXCITATION AND SENSOR PLACEMENT 

USING GENETIC ALGORITHMS 



By 
CINNAMON BUCKELS LARSON 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 

1996 



To Jim and Audrey 



. f, 



ACKNOWLEDGEMENTS 

I would like to sincerely thank my advisor, Dr. David Zimmerman, for all of the support, 
advice, and knowledge he has given to me. He has worked hard on my behalf, obtaining 
funding for my support and providing me with several research opportunities. He gave me 
the guidance to learn and the room to grow. I will forever be indebted to him. 

I would like to thank my husband, Jim, his parents, and my daughter, Audrey. Without 
their love and support I never would have made it. I would like to thank my sisters, Beth, 
Kim, Cynthia, and Erin, my brother Laing, and my best friend Leslie for their never-ending 
encouragement and love. I am truly blessed. 

I would like thank my dear friends Mohamed Kaouk and William Leath for their advice, 
support, and companionship through graduate school. I would also like to thank the entire 
staff and faculty of the Aerospace Engineering, Mechanics, and Engineering Science 
department. Specifically, I would like to acknowledge my committee. Dr. Norman Fitz-Coy, 
Dr. Daniel Drucker, Dr. Marc Hoit, Dr. Peter Ifju, and Dr. Bavani Sankar for their advice. 

I would like to acknowledge Sandia National Laboratories for their financial support 
and for the research opportunities they have given me. Specifically, I would like to thank Ed 
Marek, Clay Fulcher, and Scott Klenke. I would also like to acknowledge General Motors 
for providing data for my studies. 

I would like to thank the Florida/NASA Space Grant Consortium whose financial 
support made my graduate studies possible. 



m 



TABLE OF CONTENTS 



page 



111 



ACKNOWLEDGEMENTS 

LIST OF TABLES vii 

LIST OF FIGURES viii 

KEY TO ABBREVIATIONS xi 

ABSTRACT xii 

CHAPTERS 

1 INTRODUCTION . . . 



1 . 1 Finite Element Model Refinement 2 

1.2 Modal Testing: Sensor and Actuator Placement 5 

1 .3 Current Study Objective 7 

2 GENETIC ALGORITHMS: THEORY AND APPLICATION 9 

3 FINITE ELEMENT MODEL REFINEMENT USING 

GENETIC ALGORITHMS 14 

3.1 Introduction 14 

3.2 Model Refinement Problem Formulation 14 

3.3 Genetic Algorithm Application 15 

3.4 Numerical Example: Six Bay Truss FEM I7 

3.4. 1 Model Refinement 18 

3.4.2 Effect of Noise 20 

3.5 Conclusions 22 

4 MODAL TEST EXCITATION AND SENSOR PLACEMENT: 

CURRENT TECHNIQUES 25 



IV 



4.1 Introduction 25 

4.2 Effective Independence 25 

4.3 Kinetic Energy 29 

4.4 Eigenvector Product 31 

4.5 Driving Point Residue 31 

5 MODAL TEST EXCITATION AND SENSOR PLACEMENT: 

NEW TECHNIQUES 35 

5.1 Introduction 35 

5.2 Mode Indicator Function 35 

5.2. 1 Excitation Placement 36 

5.2.2 Sensor Placement 39 

5.3 Observability and Controllability '. 40 

5.3.1 Excitation Placement 45 

5.3.2 Sensor Placement 47 

6 PRE-MODAL TEST PLANNING ALGORITHM APPLICATION: 

NASA EIGHT-BAY TRUSS 49 

6.1 Introduction 49 

6.2 NASA Eight-Bay Test Bed 49 

6.2. 1 Excitation Placement 50 

6.2.2 Sensor Placement 54 

6.2.3 Results: Random Sensor Location 57 

6.3 Computational Efficiency 58 

6.4 Conclusion 50 

7 PRE-MODAL TEST PLANNING ALGORITHM APPLICATION: 
MICRO-PRECISION INTERFEROMETER TRUSS 61 

7.1 Introduction 61 

7.2 Micro-Precision Interferometer Test Bed 61 

7.3 Excitation Placement 62 

7.4 Sensor Placement 70 

7.4.1 Unconstrained Sensor Placement 7I 



7.4.2 Triaxially Constrained Sensor Placement 75 

7.4.3 Unconstrained vs. Triaxially-Constrained Sensor Sets go 

7.5 Effect of Model Error gO 

7.5.1 Excitation Placement with Model Error g2 

7.5.2 Sensor Placement with Model Error g3 

7.6 Computational Cost gg 

7.7 Conclusions go 

8 PRE-MODAL TEST PLANNING APPLICATION: 

CAR BODY 95 

8. 1 Introduction 95 

8.2 Excitation Placement 95 

8.3 Sensor Placement 102 

9 CONCLUSIONS AND FUTURE WORK 105 

REFERENCES ^08 

BIOGRAPHICAL SKETCH jl2 



VI 



■ fTij^ffy 



LIST OF TABLES 

Table page 

5. 1 GMEF Design Variable Description 38 

6. 1 Eight-Bay Truss Frequencies and Mode Description 50 

6.2 Percent Difference in FE and Identified Frequencies 57 

6.3 Total Floating Point Calculations for Each Placement Technique 60 

7. 1 Reduced MPI FEM Frequencies Compared with MPI 

Modal Test Frequencies 64 

7.2 Original, GMIF, and GCON Excitation Locations MIF Values 66 

7.3 Difference Between Pre- and Post-Corrupted Model 

Frequencies and Mode Shapes 82 

7.4 Number of Sensor or Triax Sets That Change 

When Model Error Is Added 87 

7.5 Floating Point Calculations for MPI Sensor and Excitation Placement 89 

8.1 Car Body Excitation Location and Orientation 96 

8.2 Mode Indicator Function Values for Various Excitation Placements 97 , 

8.3 Controllability Angles (in degrees) for Excitation Placements loi 

8.4 Excitation Placement Techniques Floating Point Calculations 10 1 

8.5 Triaxial Sensor Placement Techniques Floating Point Calculations 104 



Vll 



LIST OF FIGURES 

Figure page 

1 . 1 Finite Element Modeling 2 

1 .2 Finite Element Model Refinement 3 

1.3 Modal Testing 6 

2. 1 Genetic Algorithms as Robust Problem Solvers 9 

2.2 Coding of a Four Design Variable Problem \i 

2.3 Cross-Over Examples 12 

2.4 Genetic Algorithm Flow Chart 13 

3.1 Six Bay Truss with 25 DOFs 17 

3.2 Generational Data, Measured Modes with No Noise 19 

3.3 Generational Eigensolution Data, Measured Modes with No Noise 21 

3.4 FRF after 0, 5, 10, and 20 Generations, Measured Modes with No Noise ... 21 

3.5 Generational Data, Measured Modes with 15% Noise 23 

3.6 Generational Eigensolution Data, Measured Modes with 15% Noise 23 

3.7 FRF After 20 Generations, Measured Modes with 15% Noise 24 

4. 1 Typical Driving Point Residue (NASA 8-Bay Truss) 34 

5.1 Typical MIF Plot 37 

5.2 Excitation Selection by GMIF 38 

5.3 State Space Variable Description 42 

5.4 Controllability and Observability 44 

6.1 NASA 8-Bay Truss 50 

6.2 Eight-Bay Excitation Locations 5I 



vin 



6.3 Eight-Bay Excitation Locations Frequency Response 

of Time Domain Data 53 

6.4 Eight-Bay Excitation Placement Cross-Orthogonality of 

Identified and FEM Modes 1 to 5 53 

6.5 Eight-Bay Sensor Locations 55 

6.6 Eight-Bay Sensor Placement Cross-Orthogonahty of 

Identified and FEM Modes 1 to 5 55 

6.7 Eight-Bay Cross-Orthogonalities of Five Techniques Compared to 300 

Random Sensor Sets 59 

7.1 MPI Structure 52 

7.2 Excitation Placement on MPI Structure 53 

7.3 Typical Frequency Response for MPI Structure 65 

7.4 Comparison of Selected Excitation and Random Excitation MIF Values ... 68 

7.5 Comparison of Selected Excitation and Random Excitation 

Controllability Angles 69 

7.6 Cross-Orthogonality Between FE Modes and Identified Modes 70 

7.7 Unconstrained MPI Sensor Sets 72 

7.8 Cross-Orthogonality Between MPI FE and Identified Modes, 

18 Unconstrained Sensors 74 

7.9 Triaxially Constrained MPI Sensor Sets 77 s 

7.10 Cross-Orthogonality Between MPI FE and Identified Modes, 

6 Triaxially Constrained Sensors 7g 

7.11 Model Error Added to MPI FEM gl 

7.12 True vs. Corrupted MPI Mode Shapes gj 

7.13 GMIF Derived Excitation Locations g3 

7.14 GCON Derived Excitation Locations g4 

7.15 Model Error Effect on Unconstrained MPI Sensor Sets g5 

7.16 Model Error Effect on Triaxially Constrained MPI Sensor Sets g6 

7. 17 Cross-Orthogonality Between MPI Identified Modes Using Corrupted Model 

and Uncorrupted FEM Modes (18 Unconstrained Sensors) 91 



IX 



7.18 Cross-Orthogonality Between MPI Identified Modes Using Corrupted 

Model and Uncorrupted FEM Modes (6 Triaxially Constrained Sensors) ... 93 

8. 1 Car Body Shaker Locations 96 

8.2 MIF Values for Excitation Placements Compared to 500 Randomly 

Located 3-Point Excitations 99 

8.3 Controllability Values for Excitation Placements Compared to 500 

Randomly Located 3-Point Excitations 100 

8.4 Triaxially Located Car Body Sensor Sets 102 



KEY TO ABBREVIATIONS 



1'-, 



AKE average kinetic energy c 

ARS average random sample , ; , 

DOF degree of freedom - %,-* 

DPR driving-point residue 

EI effective independence 

ERA eigensystem realization algorithm 

EVP eigenvector product 

FEM finite element model 

FRF frequency response function 

GA genetic algorithm > 

GCON .... genetic controllability algorithm 

GMIF genetic mode indicator function algorithm 

GMRA .... genetic model refinement algorithm 

KE kinetic energy 

MIF mode indicator function 

MPI micro-precision interferometer 

OBS observabihty 



XI 



Abstract of Dissertation Presented to the Graduate School 

of the University of Florida in Partial Fulfillment of the 

Requirement for the Degree of Doctor of Philosophy 

VIBRATION TESTING BY DESIGN: 

EXCITATION AND SENSOR PLACEMENT 

USING GENETIC ALGORITHMS 

By 

Cinnamon Buckels Larson 
May 1996 

Chairperson: Dr. David C. Zimmerman 

Major Department: Aerospace Engineering, Mechanics and Engineering Science 

This dissertation is an investigation of the use of genetic algorithms for the purposes of 
finite element model refinement and pre-modal test planning. The objective of a model 
refinement technique is to use information, about a structure, obtained during a vibration test 
to update the analytical model. The product of this process is an updated model of a structure 
which possesses dynamic properties closer to the dynamics obtained from the modal test of 
the structure. A genetic algorithm is used to vary finite element structural parameters to 
obtain an updated model with measured modal properties. 

Although one purpose of a modal test is to use the information to update finite element 
models, the information obtained may be used for other purposes such as damage 
assessment, critical loads and frequency determination, and vibration control design. The 
type of information to be realized from a vibration test may well govern how and where the 
structure is excited and observed. The principal purpose of the current work is to explore the 
subject of pre-modal test planning for excitation and sensor placement. An overview of 
several existing sensor and excitation placement techniques is presented as a platform for the 
current study. The sensor placement techniques include effective independence, kinetic 



Xll 



energy, and eigenvector product and the excitation placement techniques include 
eigenvector product, kinetic energy, and driving-point residue. Two new sensor and two new 
excitation placement techniques are developed using normal mode indicator functions, and 
the concept of modal controllability and observability along with genetic algorithms. The 
new and existing techniques are compared using three finite element models: the NASA 
eight-bay truss, the Jet Propulsion Laboratory Micro-Precision Interferometer test bed, and a 
car body. 



xni 



CHAPTER 1 
INTRODUCTION 



The area of structural engineering encompasses the design, manufacture, and test of a 
wide variety of systems. These basic steps are all used in the design of any structure, whether 
it is a household appliance, an automobile, a bridge, an aircraft, or a spacecraft. In the past, 
systems were over-engineered and over-built resulting in an increase in the time and material 
required to build them. Often times the steps in the engineering process were repeated 
several times until the designed system performed satisfactorily. With the advent of the 
computer, tools have been and are continuing to be developed which enable the structural 
engineer to improve on each step of the engineering process. These tools not only help to 
limit the time, material, and cost that it takes to complete the engineering steps but they also 
help to limit the repetition of these steps. 

Structural computer modelling and vibration testing are two tools that have been 
developed with the aid of computers. Knowledge about material properties and structural 
dynamics may be used to create computer models of a system, which in turn may be used to 
predict dynamic performance and limitations. Once the system has been built, modal testing 
may be used to gain a greater insight into the dynamics of the structure and to update the 
computer model. An updated computer model may be used as a health monitoring tool for 
the structure after it goes into use. 



1.1 Finite Element Model ReFinement 



One of the most common modelling techniques is finite element modelling (FEM) 
which can be used to represent the continuous medium of a structure as a connection of finite 
elements. This enables a system with distributed mass, damping, and stiffness properties to 
be represented as a lumped parameter system with discrete mass damping and stiffness 
properties (Figure 1.1). In other words, an infinite degree of freedom system is represented 
as a finite degree of freedom system. 





/ FINITE ELEMENT MODEL 



^21 



ZZl 



I \ \ \ 



3; 



^ 




Figure 1 . 1 Finite Element Modeling 

A wide variety of finite element modelling software is available to the designer. The basic 
steps involved in the modelling process are as follows: 

1 . Divide continuum into a finite number of elements. 

2. Select node points where equilibrium conditions are enforced. 

3. Determine element types and properties. 

- elemental type (rod, plate, etc.) and location 

- elemental displacement, stiffness, stress-strain, node-point lodes 

4. Assemble elemental matrices (mass,stiffness, and damping). 



5. Develop equilibrium equations for node point location. 



6. Create global mass damping and stiffness matrices from which frequencies and 
mode shapes are calculated. 

The resulting FEM may be used to evaluate the efficacy of a design before it is built. 
Critical loads, resonant frequencies, and mode shapes may be predicted using the FEM and 
appropriate changes to the design based on these values may be made. Once the structure has 
been built, the FEM may be used to predict the performance of the structure under working 
conditions as well as serve as a damage assessment tool. However, while the original FEM is 
a predicted representation of a particular structure, the dynamic performance of the FEM 
very rarely matches the performance of the as-built structure. In order to correlate the 
dynamics of the structure to those of the FEM, the model must undergo a refinement process. 
The basic steps of FEM refinement are shown in Figure 1.2. The dynamic properties of the 
FEM are compared to the dynamic properties extracted from a vibration test of the structure. 
The resulting information is used to refine the model so that the modal properties of the FEM 
agree with modal properties from the vibration test. 




MODAL ANALYSIS 

frequencies 
mode shapes 



REFINED 

FINITE ELEMENT 

MODEL 




Figure 1 .2 Finite Element Model Refinement 



Some of the earliest work done in model refinement was proposed by Rodden (1967), 
who explored using modal test data to generate analytical mass and stiffness properties of the 
structure being tested. The early work of Rodden has broadened into the modern FEM 
refinement techniques. Algorithms used to address the FEM refinement can be broadly 
classified as falling into one of four different approaches: optimal-matrix updates, sensitivity 
methods, eigenstructure assignment techniques, and minimum-rank perturbation methods. 
Survey papers providing an overview of these techniques are provided in papers by Ibrahim 
and Saafan, 1987; Heylen and Sas, 1987; and Zimmerman and Kaouk, 1992. 

In the optimal matrix update formulation, perturbation matrices for the mass, stiffness, 
and/or damping matrices are determined which minimize a given cost function subject to 
various constraints. Typical constraints may include satisfaction of the eigenproblem for all 
measured modes, definiteness of the updated property matrices and preservation of the 
original sparsity pattern of the property matrices. Baruch and Bar Itzhack (1978) worked on 
an optimal update of the global stiffness matrix with a cost function that minimized the 
Frobenius norm of the perturbation matrix. Their work was expanded to look at updating 
mass, damping, and stiffness matrices (Berman and Nagy, 1983; Fuh et al., 1984; Hanagudet 
al., 1984). Kabe (1985) and Kammer (1987) expanded on this work further by looking at 
matrix updating while preserving the sparsity pattern of the original FE global matrices. 

Sensitivity methods for model refinement make use of sensitivity derivatives of modal 
parameters with respect to physical design variables (Martinez et al . , 1 99 1 ) or with respect to 
matrix element variables (Matzen, 1987). When varying physical parameters, the updated 
model is consistent within the original FE program framework. A variety of derivatives and 
optimization techniques have been used (Collins et al., 1974; Chen and Garba, 1980; 
Adelman and Haftka, 1986; Creamer and Hendricks, 1987; Flanigan, 1991). In the current 
work a physical parameter update technique using a genetic algorithm is developed. 

Inman and Minas (1990) proposed designing pseudo-controllers to be applied to the 
FEM in an iterative fashion resulting in a match between measured and FE modal properties. 



These controllers were then translated into matrix updates. These techniques, known as 
control-based eigenstructure assignment techniques, are based on work done in 
eigenstructure control (Andry et al., 1983). Zimmerman and Widengren (1990) proposed a 
non-iterative eigenstructure assignment formulation using an algebraic Riccati equation. 

Finally, the development of a minimum rank update theory has been recently proposed 
as a computationally attractive approach for model refinement and damage detection 
(Zimmerman and Kaouk, 1992). The update to each property matrix is of minimum rank and 
is equal to the number of experimentally measured modes which the modified model is to 
match. 

1.2 Modal Testing: Sensor and Actuator Placement 

Regardless of the method used to perform FEM refinement, a modal test must be 
performed on a structure or its components in order to obtain the experimental information to 
correlate with the analytical information contained in the FEM. Finite element model 
refinement is only one use for modal data. Modal analysis is also a tool for damage 
assessment and force reconstruction. Several issues may govern the use of modal tests for 
these purposes. The final use of modal test data governs the pretest planning associated with 
modal testing. The placement of actuators for structural excitation purposes and the 
placement of sensors for structural response observations may well depend on whether the 
data will be used for modal parameter estimation, mode orthogonality for FEM correlation, 
identification of uncertain parameters in FEMs, structural health monitoring, or force 
reconstruction. 

The science of modal testing is thoroughly discussed in D.J Ewins' book Modal Testing: 
Theory and Practice. The basic steps involved in a modal test are discussed here for 
completeness and are pictured in Figure 1.3. The structure being tested must be dynamically 
excited and the response of the structure to this input must be measured. The excitation may 
be accomplished using an impact hammer, a shaker, or a release from an initial structural 



displacement. The response of the structure is generally measured using piezoelectric 
accelerometers, which are mounted on the structure in various locations. The force and 
response signals are sent to a processor or analyzer, after being filtered and amplified, from 
which a frequency response function (FRF) of the structure is obtained. The modal 
properties (mode shapes, damping ratios, and frequencies) may be obtained from the 
analysis of the FRF. 



FORCE TRANSDUCER 



structure 



RESPONSE TRANSDUCER 



triax 



single DOF 




conditioning 
amplifiers 



frequencies 
mode shapes 




signal processor 



Figure 1.3 Modal Testing 



Modal testing of a structure can be a costly venture in terms of time and money. 
Pre-modal test planning can be essential in saving cost by determining ahead of time the 
appropriate transducers and analyzers to use for the job at hand. Another aspect of pre-test 
planning, the one associated with the current study, is the optimal selection of the location of 
the force and response transducers. It is desired to obtain the most information at the least 
cost, which involves minimizing the number of transducers to be used. The selection of the 



transducers will also depend on what the modal data will be used for (i.e., model refinement 
or damage detection). 

A majority of the sensor placement research that has been done may be broadly 
classified into two areas, system identification and optimal control. Yu and Seinfield (1973), 
Le Pourhiet and Le Letty (1978), Omatu et al. (1978), Sawaragi et al. (1978), and Qureshi et 
al. ( 1 980) have all done work in the area of sensor placement for system identification. Shah 
and Udwadia ( 1 978) and Udwadia and Garba (1985) have done work in sensor placement for 
structural parametric identification. Kammer (1991) approached the system identification 
sensor placement problem for the purpose of FEM validation. Goodson and Polls (1978) 
researched the selection of sensors for optimal structural control. Juang and Rodriguez 
(1979) looked at sensor placement for identification and control purposes. 

Information about the FEM has been used by modal test designers to place sensors and 
actuators on a structure for the purpose of modal testing. Finite element DOFs which have 
high kinetic energy are good choices for sensor or actuator placement because more 
information may be extracted about and more energy may be input to the structure at these 
points (Kammer, 199 l;Flanigan and Hunt, 1993; Lim, 1991). Jarvis (1991) proposes using 
FE mode shape products to find sensor and actuator locations for modal testing. Kientzy et 
al. (1989) used driving point residues or modal participation factors to determine modal test 
excitation locations. Mode indicator functions have been used by the modal test engineers to 
tune modes during a modal test (Hunt et al., 1984). 

1.3 Current Study Objective 

Proper modal test planning is needed in order to obtain the largest amount of 
information about a structure relative to the task of the data at the smallest cost. The 
refinement of finite element models is a task which benefits from modal test planning. The 
objective of the current studies is to explore the areas of finite element model refinement and 



8 



pre-modal test planning. Specifically, the use of the optimization technique, genetic 
algorithms (GAs), in these two areas will be examined. 

The genetic algorithm is an optimization tool which has been developed in the past 20 
years. An overview of the theory and applications of the GA is given in Chapter 2. A 
structural-parameter update model-refinement algorithm is developed using a genetic 
algorithm in Chapter 3 and is applied using a FEM of a two-dimensional truss structure. 

The topics of pre-modal test planning actuator and sensor placement are examined in 
Chapters 4 and 5. In Chapter 4 several techniques which have been developed in existing 
literature are outlined. The excitation placement techniques of kinetic energy eigenvector 
product, and driving point residue and the sensor placement techniques of effective 
independence, kinetic energy, and eigenvector product are reviewed. In an effort to improve 
on the current sensor and actuator placement technologies, two new actuator and sensor 
placement techniques are developed in Chapter 5. The normal mode indicator function 
(MIF) of a FEM is used with a G A to optimally find excitation locations . In addition the MIF 
is used as a tool to locate sensors. The second sensor and actuator placement algorithms use a 
degree of controllability and observability calculated using the FEM information. 

The effectiveness of the current sensor and actuator placement techniques and those 
developed in Chapter 5 are explored using several structural FEMs in Chapters 6,7, and 8. In 
Chapter 6 NASA's Eight-Bay truss, in Chapter 7 the micro-precision interferometer truss, 
and in Chapter 8 a FEM of a General Motors car body are used as examples to explore the 
effectiveness of all of the sensor and actuator placement algorithms for the purpose of 
pre-modal test planning. Concluding remarks and a discussion of future work in the areas 
outlined above are given in Chapter 9. 



CHAPTER 2 

GENETIC ALGORITHMS: 

THEORY AND APPLICATION 



The motivation behind the development of GAs is that they are robust problem solvers 
for a wide class of problems, as depicted in Figure 2. 1 . However, it should be noted that they 
are not as efficient as nonlinear optimization techniques over the class of problems which are 
ideally suited for nonlinear optimization; namely continuous design variables with a 
continuous differentiable unimodal design space. Genetic Algorithms have the capability to 
solve continuous, discrete and a combination of continuous and discrete optimization 
problems. 

Nonlinear Optimization 
Genetic Algorithms 



O 

C 

<U 




Random Walk 



Problem Class 
Figure 2. 1 Genetic Algorithms as Robust Problem Solvers 

Genetic algorithms are an optimization method which is based on Charles Darwin's 
survival of the fittest theories (Holland, 1975). The basic concept of the GA is that a 
population of designs is allowed to evolve over a period of time. The most fit members of 
that population are most likely to survive thus enabling their genetic code (or design 
information) to be passed down to future generations. More than just the information 
contained in the initial population may be passed down to future generations. As in nature as 
the evolutionary process progresses, mutations may occur in the offspring which may or may 



10 



not result in more fit population members. Ideally, this evolutionary process will result in a 
population of members more fit than the original initial population. Genetic Algorithms may 
therefore be described as a directed random search or as a compromise between determinism 
and chance. 

Genetic algorithms are radically different from the more traditional design optimization 
techniques. Genetic algorithms work with a coding of the design variables, as opposed to 
working with the design variables directly. The search is conducted from a population of 
designs (i.e., from a large number of points in the design space), unlike the traditional 
algorithms which search from a single design point. The GA requires only objective 
function information, as opposed to gradient or other auxiliary information, which is usually 
required in other optimization techniques. The GA is based on probabilistic transition rules, 
as opposed to deterministic rules. There are five main operations in a basic GA: coding, 
evaluation, selection, crossover and mutation. 

Coding is the process in which each design variable is coded as a q-bit binary number. 
Discrete variables would each be assigned a unique binary string. A continuous design 
variable Bj is approximated by 21 discrete numbers between lower and upper bounds for the 
design variable, 

binary# _ (2.I) 

1 imin 21—1 ^""2X "imin/ ^ ^ 

where Bimin and Bi^ax are the lower and upper bounds on the i'*' continuous design variable 
and binary* is an integer number between zero and 21 -1 . The continuous to discrete coding 
is like that of an analog to digital converter used in control systems. A population member 
is obtained by concatenating all design variables to obtain a single string of ones and zeros. 
Thus, a population member contains all information to completely specify the total design. 
For example, consider a design which has three continuous variables Bi, B2 and B3 
represented by 5-bit, 6-bit, and 4-bit numbers and a discrete design variable B4 which can 
take on four different values. An example of a population of members containing this design 



11 



information is pictured in Figure 2.2. A population is defined to be a grouping of npop 
members, where npop is the number of members in the population. 





Design Variables 


population 
member 


5 bit 
continuous 
Bi 


6 bit 

continuous 

B2 


4 bit 
continuous 
B3 


2 bit 
discrete 

B4 


1 


10 111 


10 1 


11 


1 1 


2 


110 


1110 10 


110 1 


1 


• 




• 


• 


• 


npop 


110 10 


1 10 


11 


1 



Figure 2.2 Coding of a Four Design Variable Problem 

Evaluation is the process of assigning a fitness measure to each member of the current 
population. The fitness measure is typically chosen to be related to the objective function 
which is to be minimized or maximized. No gradient or auxiliary information is used; only 
the value of the fitness function is needed. Therefore, GAs are less likely than traditional 
"hill climbing" algorithms to become "trapped" at a local minima or maxima. Additionally, 
because no gradient information is required, the design space is allowed to be discontinuous. 

Selection is the operation of choosing members of the current generation to produce the 
prodigy of the next generation. Selection is biased toward the most fit members of the 
population. Therefore, designs which are better as viewed from the fitness function, and 
therefore the objective function, are more likely to be chosen as parents. 

Crossover is the process in which design information is transferred to the prodigy from 
the parents. Crossover amounts to a swapping of various strings of ones and zeroes between 
the two parents to obtain two children. Two possible types of cross-over are illustrated in 
Figure 2.3. 



"■■(,•.•* 



12 



Point Cross-over 



Pattern Cross-over 



parent 1 


101:0001 


10 10 1 


parent 2 


111:0100 


1 10 10 


child 1 


swap: point 
101:0100 


swap X X - - X - X 
pattern 

1110 10 


child 2 


1 1 1:0001 


10 1 



Figure 2.3 Cross-Over Examples 



Mutation is a low probability random operation that may perturb the design represented 
by the prodigy. The mutation operator is used to retain design information over the entire 
domain of the design space during the evolutionary process. 

Holland (1975) developed the concept of schema, which for a very simple GA 
implementation explains why GAs work. Schema are a similarity template defined by O's, 
I's and x's, where x's are the don't care symbol. Thus, for a design coded with a total of 
8-bits, one schema would be lOxxxxxx. All designs which have a 1 in the most significant 
bit and a in the 2nd most significant bit would be said to contain this schema. Holland's 
Schema Convergence Theorem states that under certain combinations of selection, 
crossover, and mutation, the expected number of schema H at generation k+1, n(H,k+l), is 
given as 



n(H,k+l)>(l-E)^n(H,k) 

Illavg 



(2.2) 



where e is a number much less than 1 and fitn and fitavg are the average fitness of all designs 
containing schema H and of the population as a whole, respectively. Thus, if those designs 
which contain schema H have on the whole a higher average fitness than the overall general 
population, the expected number of schema H in the next generation will be greater than or 
equal to the number of schema H in the current generation. 



13 



The proof is valid only for a specific combination of selection, crossover and mutation. 
It should be noted that a general proof for more complex GAs has not been developed. 
However, there exists a wide body of literature which demonstrates the power and capability 
of advanced GAs (Schaffer 1989, Grefenstette 1987). A flow chart summarizing the GA 
process is shown in Figure 2.4. - 



CODING 

Create Initial Population 



EVALUATION 

Evalute Population Fitness 
(objective function calculation) 



— SELECTION 

Apply Selection Criteria 
(which members reproduce?) 



CROSSOVER and MUTATION 
create new members 



NO 




•" New Design 



Figure 2.4 Genetic Algorithm Flow Chart 



CHAPTER 3 

FINITE ELEMENT MODEL REFINEMENT 

USING GENETIC ALGORITHMS 



3.1 Introduction 

As discussed in Chapter 1 , an important tool in the design of engineering structures is the 
finite element model (FEM). Recall that FEM refinement techniques may be classified as 
optimal matrix updates, sensitivity updates, eigenstructure assignment updates, and 
minimum-rank perturbation updates. Sensitivity methods use sensitivity derivatives of 
modal parameters with respect to physical design variables or with respect to matrix element 
variables. These derivatives are used in order to determine what changes to make to the 
physical parameters or elemental matrices of the FEM in order to obtain in a refined model 
with measured modal properties. A GA-driven model refinement technique is developed to 
update FE structural parameters to provide an updated model with the measured modal 
characteristics. This model refinement technique is illustrated using a numerical example. 

3.2 Model Refinement Problem Formulation 

For a given undamped structure it is assumed that an n-DOF FEM is developed and 
results in the second-order linear differential equation of motion, 

Mx -I- Kx = (3.2.1) 

where M and K are the original analytic (nxn) mass and stiffness matrices, and x is the («xl) 
position vector. The over-dots represent differentiation with respect to time. The eigenvalue 
problem associated with Eq. (3.2.1) can be written as 



14 



15 



l^M^^ + % = (3.2.2) 

where A^ is the r^^ eigenvalue and ^r is the r* mass orthogonal eigenvector of the original 
analytical system. It is assumed that the original analytic model of Eq. (3.2.1) does not 
satisfy the eigenvalues (A^r) and the mass orthogonal eigenvectors (^^r) of the 
experimentally measured system, 

>^mrMm^^j + K^^^^ = (3.2.3) 

where Mm and Kn, are the experimentally derived mass and stiffness matrices of the 
structure. Therefore, a discrepancy between the original analytic and measured modal 
information will result in an eigenvalue problem of the form 

IUM + AM(p)]^^^ + [K + AK(p)]^^^ = (3.2.4) 

where AM(p) and AK(g) are perturbation matrices which are functions of the structural 
parameters vector p. These perturbation matrices represent the mismatch between the 
original analytic mass and stiffness matrices and the experimentally derived mass and 
stiffness matrices. In order to develop an updated analytical model which is in agreement 
with measured modal data, a structural parameters vector p must be found which satisfies 
Eq. (3.2.4) for all measured modes. In the following sections, a model refinement technique 
is developed which employes the use of a GA to find the structural parameters vector which 
will result in an updated FEM whose modal properties match the measured modal properties 
of the as-built structure. 

3.3 Genetic Algorithm Application 

As explained in Chapter 1 , a FEM is a lumped parameter representation of a continuous 
structure. Information about the geometry and material of a structure are used to estimate its 
properties and to create the FEM. Mass, density. Young's modulus, cross-secdonal area, and 
moment of inertia are some example of the properties which are used to develop a FEM. 



16 



Since these properties are estimated for the structure, they may be perturbed to give an 
updated FEM with the same modal properties as experimentally measured from the true 
structure. A Genetic Model Refinement Algorithm (GMRA) is developed which uses a GA 
to search for updated structural parameters which will result in an updated FEM with 
corresponding measured modal properties. An outline of the steps of GMRA, which follow 
those for a GA given in Chapter 2, follows. 

Coding . The design variables used in GMRA are continuous design variables which 
represent the structural parameters to be changed. Limits may be set on the amount of 
perturbation allowed for each design variable or structural parameter. This enables the user 
to allow small perturbations to variables about which they are certain, such as cross-sectional 
area or moment of inertia and larger perturbations to variables about which they are less 
certain, such as density or Young's modulus. 

Evaluation . The most fit members of a population are those which minimize a chosen 
objective function. An objective function has been formulated which states, the most fit 
structural parameters vector p is one which results in an updated analytical model which 
gives the smallest value for the objective function 

J.«=iS^"'+ill*»r-$„„l|l'r (3.3.1) 

r= 1 r= 1 

where X^jr and ^ur ^e the r'^ eigenvalue and eigenvector of the updated analytical model. 
The first summation of Eq. (3.3.1) provides for the minimization between measured 
eigenvalues and updated analytic eigenvalues. As the updated eigenvalue approaches the 
measured eigenvalue the first summation approaches zero. The absolute value of the 
difference in each measured and updated eigenvalue is divided by the corresponding 
measured eigenvalue to insure that each frequency contributes equally to the objective 
function. The second summation of Eq. (3.3.1) is the 2-norm between the difference in 
measured and updated eigenvectors and provides for the minimization between measured 
and updated mode shapes. As the updated eigenvector approaches the measured eigenvector 



17 



the second summation approaches zero. The variables Wr and hp are weights which can be 
changed in order to emphasize agreement between specific measured and updated 
eigenvalues/eigenvectors. By changing the weights, emphasis can be placed either on 
updated eigenvalue or updated eigenvector agreement with measured data. 



3.4 Numerical Example: Six Bay Truss FEM 

A finite element model of a six bay truss with 25-DOFs was developed to test GMRA. A 
picture of the truss is given in Figure 3.1. 



A= Ucm^ 
E = 7.03xl09kg/m2 
p = 2685 kg/m3 
/ = 0.75 m 




Figure 3.1 Six Bay Truss With 25 DOFs 



In Figure 3. 1 , A is the cross sectional area , E is the modulus of elasticity, p is the density 
of all of the members, and / is the length of each bay. It is assumed that the analytic model is 
incorrect and needs to be changed in order to facilitate agreement in analytic and measured 
modal properties. To obtain experimental modal information, the FEM of the six bay truss is 
altered and the "experimental" modal information is calculated. It is assumed that the 
dimension of the measured eigenvector is the same as that of the analytic eigenvector. This 
can be accomplished using an eigenvector expansion algorithm (Herman and Nagy, 1971; 
Smith and Beatie,1990; Zimmerman and Smith, 1992). An alternate formulation would be 
to let the vector norm calculation of Eq. (3.3. 1) take place over only those components of the 
eigenvector that are measured, therefore, eliminating any error that may be introduced by 
expanding the measured eigenvectors. 



18 



3.4.1 Model Refinement 

As a first example, it is assumed that all of the structural parameters are known to be 
correct except for E, the modulus of elasticity. In addition, the properties of all of the 
diagonal members, all of the horizontal members, and all of the vertical members are linked. 
GMRA is used to find the updated structural parameters vector, pu> which minimizes Eq. 
(3.3.1), 

Pu = -fEd Eh Ev} (3.4.1) 

The components of Pu (Ed> Eh, and Ey ) are the moduli of elasticity of the diagonal, 
horizontal, and vertical truss members. The original analytic model has the same modulus 
of elasticity for all members, which will be referred to as Enom- The resulting analytical 
parameters vector is 

P^ ~ lEnom Enom '^nomf (3.4.2) 

The value for Enom is 7.03x10^ kg/m^. The structural parameters vector which is used to 
generate the experimentally measured model is 

p^ = {0.95Enom 0.90Enom 0.92E„om} (3.4.3) 

The genetic algorithm is instructed to search for three design variables (components of 
the structural parameters vector) which are in the range of 0.7Enom to 1.3Enom- An initial 
population representing the horizontal, vertical, and diagonal moduli of elasticity is 
randomly generated within the limits of O.TEnom to 1.3Enom. A member is added to this 
initial population which represents the original analytic model parameters vector. Since the 
original population is randomly generated, there is a good possibility that one of those initial 
population members will be more fit than the member representing the original analytic 
model. Therefore, the improvement in the updated eigensolution would in part be due to a 
random search. Even though this random chance is a benefit of genetics, in order to show the 
true improvement to the original analytic model by the genetic algorithm, an initial 



19 



population is chosen with all members less fit than the member representing the original 
analytic model. In order to facilitate frequency matching, the weights of Eq. (3.3.1) are set to 
emphasize minimization of the eigenvalue portion of the cost function. Five measured 
modes are suppUed and GMRA is instructed to run for twenty generations (Figure 3.2). 



My iimum and Average Objective Functi ons 



c 

u 
> 

•a 

u 
u 




c 

E 
o 
c 

u 



10 20 

Generation 



o 
c 

u 

(4-1 

o 

I 

a 

a. 





Horizontal Truss Members 


1.2 
1 


A 


0.8 





o 

c 



10 
Generation 



20 



Diagonal Truss Members 




10 
Generation 

Vertical Truss Members 




10 
Generation 



^■^; 



Figure 3.2 Generational Data, Measured Modes with No Noise 



The top left graph of Figure 3.2 shows the average fitness (dashed line) and maximum 
fitness (solid line) of the current population at each generation. As the generations increase 
the average and maximum fitness improves, which corresponds to a decrease in the value of 
the objective function. After the first generation there is substantial improvement in the 
updated objective function over the original analytic objective function value. Since the 
members of the population which are randomly generated are all less fit than the original 
analytical model member, the decrease in objective function over the first generation is due 
to the children of the initial population. Since it is the goal to minimize the objective 
function, this is a desirable trend. Also as the generation increases, the average fitness 
approaches the maximum fitness. This is due to the fact that as generations evolve, the 



20 



overall population tends toward the most fit member. After a certain number of generations, 
the diversity in the population members decreases. 

The other three graphs of Figure 3.2 show how the actual design variables (moduli of 
elasticity) are varying over the generations. The straight lines on these graphs correspond to 
the "experimental" diagonal, horizontal, and vertical moduli of elasticity which were used to 
generate the measured modes. It is seen that within the first few generations, the parameters 
have quickly converged near their "experimental" values. In later generations it is seen that 
Ey, the vertical member's elastic modulus, varies widely. Physically, this is due to the fact the 
lower modes are fairly insensitive to the stiffness of the vertical members. 

The generational trends of the eigensolution of the updated model are shown in Figure 
3.3. The top left graph is the norm of the difference in the measured and updated 
eigenvectors. It can be seen that over some of the generations the value of this norm increases 
instead of decreasing. This is due to the fact that the eigenvalue portion of the cost function is 
weighted more heavily in order to insure modal frequency agreement. It can be seen that an 
increase in the norm in Figure 3.3 corresponds to the overall updated eigenvalues moving 
closer to the measured eigenvalues resulting in a decrease in the value of the objective 
function. A comparison of the first three frequencies of the updated model with respect to the 
measured frequencies over 20 generations is shown in the other three graphs of Figure 3.3. 

The FRFs of the pre-genetics model and the FRFs of the post-genetic models after 0, 5, 
10, and 20 generations are pictured in Figure 3.4. An immediate improvement in the FRF of 
the system can be seen after five generations. 

3.4.2 Effect of Noise 

The example presented would be expected to behave differently when there is noise 
present in the measured modal information. To simulate noise in the measured data, 5% and 
15% random noise were added to the measured eigenvectors. It is assumed that the modal 
frequencies are measured accurately. As in the case with no noise, an initial population is 



21 



N orm (Measured-Updated) Eigenvecto rs j^ rst Frequency: Measured(..) Updated (-) 



o 




10 20 

Generation 




10 20 

Generation 



Sec ond Frequency: Measured(..) Update d(-) Th ird Frequency: Measured(..) Updated (-) 




Generation 



Generation 



Figure 3.3 Generational Eigensolution Data, Measured Modes with No Noise 



After Generations 



After 5 Generations 




100 200 

Frequency (Hz) 

After 10 Generations 



100 200 

Frequency (Hz) 

After 20 Generations 



100 200 

Frequency (Hz) 



100 200 

Frequency (Hz) 



300 




300 



(...) Updated Model (-) Measured Model 
Figure 3.4 FRF After 0, 5, 10, and 20 Generations, Measured Modes with No Noise 



22 



generated with one member which represents the original analytic model and with other 
members randomly generated which have cost function values greater than that for the 
original analytic model member. This is to show how the cost function is minimized due to 
genetics and not j ust due to a random selection of a more fit design. GMRA was run using 5 % 
and 15% noise with 5 measured modes. The generational results for 15% noise are pictured 
in Figures 3.5 and 3.6, and the FRF is pictured in Figure 3.7. Graphically a similar trend was 
observed for 5% noise. 

An immediate improvement in the cost function can be seen after 5 generations, as 
shown in the top left graph of Figure 3.5. The generational updated eigensolution data of 
Figure 3.6 shows a similar trend to that of the example with no noise. The norm of the 
difference in measured eigenvectors with 15% noise and the updated eigenvectors is given in 
the top left graph of Figure 3.6. The straight dotted line in this figure corresponds to the norm 
of the difference in measured eigenvectors with and without noise. 

The FRF pictured in Figure 3.7 shows how the model which was updated using noisy 
data compares with measured model without noise and with the original analytic model. 
Because the cost function was weighted to be more heavily affected by the frequencies, the 
effect that noisy modes may have had on the update was minimized. 

3.5 Conclusions 

One of the benefits of using a GA to search for an updated parameters vector is that the 
search is conducted from several points in the design space whereas conventional gradient 
sensitivity methods conduct the search from a single point. This helps enable GMRA to 
avoid getting stuck in a local minimum in addition to completing the search faster. Based on 
the evaluation of the data of this example, GMRA was successful in identifying an updated 
parameters vector which resulted in an updated FEM with measured modal properties. One, 
draw back to GMRA is that it requires an eigensolution of the FEM in order to calculate the 
objective function of Eq. (3.3.1). For large FEMs this objective function evaluation is a 






23 



computationally expensive calculation, and would need to be redesigned to make GMRA a 
feasible model refinement tool. 



MJJ iimum and Average Objective Functi ons 



c 



(U 

'S 
O 



u 



c 
o 



a. 




10 
Generation 

Horizontal Truss Members 



10 
Generation 



o 
c 




o 

c 

u 

o 
c 
o 

o 



Diagonal Truss Members 




10 20 

Generation 

Vertical Truss Members 




10 20 

Generation 



Figure 3.5 Generational Data, Measured Modes with 15% Noise 



l^ orm (Measured-Updated) Eigenvecto rs ^ rst Frequency: Measured(..) IJpdated (-) 




3 

a- 



10 20 

Generation 




10 
Generation 



ond Frequency: Measured(..) Update d(-) Th ird Frequency: Measured(..) Updated (-) 







3 

a- 




10 20 

Generation Generation 

Figure 3.6 Generational Eigensolution Data, Measured Modes with 15% Noise 



24 



Analytic Model(— ), Measured Model No Noise(-), and Updated Model (...) 




100 150 200 

Frequency (Hz) 



300 



Figure 3.7 FRF After 20 Generations, Measured Modes with 15% Noise 



CHAPTER 4 

MODAL TEST EXCITATION AND SENSOR PLACEMENT: 

CURRENT TECHNIQUES 



4.1 Introduction 

In the current literature various techniques for excitation and sensor selection for modal 
testing exist. These techniques vary in computational complexity, cost, and accuracy. 
Several of these techniques were explored in the current study as a basis for comparison for 
the excitation and sensor placement techniques developed in the next chapter. An overview 
of the excitation placement techniques of kinetic energy, driving point residues, and 
eigenvector products and of the sensor placement techniques of effective independence, 
kinetic energy, and eigenvector product is given in the following sections. 

4.2 Effective Independence 

Effective independence (EI) is a technique developed to place sensors for the purpose of 
obtaining structural information for FEM verification for large space structures (Kammer, 
1991). It follows from the work done by Shah and Udwadia (1978) and Udwadia and Garba 
(1985). The sensor locations are chosen such that the trace and determinant of the Fisher 
information matrix (corresponding to the target modal partitions) are maximized and the 
condition number minimized. By maximizing the determinant of the Fisher information 
matrix, the covariance matrix of the estimate error would be minimized, thus giving the best 
estimate of the structural response. A reduced sensor set is obtained in an iterative fashion 
from an initial candidate set by removing sensors from those DOFs (i.e., removing rows from 
the Fisher information matrix) which contribute least to the linear independence of the target 
modes. 



25 



26 



In order to perform test analysis mode shape correlation using a cross-orthogonality 
criterion, the measured modes obtained during the modal test must be linearly independent. 
A summary of the derivation given in Kammer's paper (1991) follows. The output of the 
sensors can be expressed as the product between the FEM target mode matrix partitioned to a 
candidate set of sensors, Os, and the modal coordinates q 

Us = Osq + W2 = H + ^2 (4.2.1) 

with Gaussian white noise ^q^ added. It is assumed that the FEM mode shapes are linearly 
independent. The sensors are sampled and an estimate of the state of the system is calculated 
as 



q = 



<DT(D 



s ^s 



^in. 



(4.2.2) 



In order to obtain the best estimate of the state of the structure, the covariance matrix of the 
estimate error must be a minimum. The covariance matrix is given by 



P = E[(q - q)(q - q)^ 



6H 



n 



-1/ 



6H 

6q> 



-1 



(4.2.3) 



Assuming that the sensors measure displacement (acceleration may also be considered), the 
covariance matrix may be rewritten as 



P = 



<I>7(W2) O, 



-1 



= Q 



-1 



(4.2.4) 



where Q is the Fisher information matrix and can be rewritten as 



Q = ^ojo, = J5A0 



(4.2.5) 



Aq will now be referred to as the Fisher information matrix. In order to minimize the 
covariance error P, Q must be maximized; therefore Aq must be maximized. Kammer ( 1 99 1 ) 
states that the determinant of the Fisher information matrix for the best linear estimate is a 
maximum for all linear unbiased estimators. Therefore, one wishes to maximize the 



27 



determinant of the Fisher information matrix. From Eq. (4.2.5), the Fisher information 
matrix is calculated to be the product of the transpose of the target mode matrix times the 
target mode matrix, 

Ao = OjOs (4.2.6) 

The first step is to calculate the eigenvalues Xp, and eigenvectors V|/a of the Fisher 
information matrix. Since it is assumed that the original FEM mode shapes are linearly 
independent, Aq will be positive definite, the eigenvalues will be real and positive and the 
eigenvectors will be orthonormal. The next step is to form the matrix product, 

G = [OsW^] [Os^a] (4.2.7) 

where is an element by element multiplication. The columns of G sum up to be the 
eigenvalues of Aq. Next the G matrix is scaled by the inverse of the eigenvalues of Aq, 



Fe = G 



\ 



^A^ 



(4.2.8) 



The effective independence vector is then calculated by summing the rows of the Fg matrix. 



n 

If 



Elj 



j = l 
n 



If 



E2j 



Ed = < 



j = i 



X. FE(nDOF)j 
j = l 



(4.2.9) 



where n is the number of target modes. The i* term in the Eq vector is hypothesized to be 
the contribution of the i^^ sensor to the linear independence of the FEM modes. A value of 
1.0 in the Eq vector corresponds to a DOF that is essential to the linear independence of the 
target modes (i.e., that DOF must be retained as a measurement location). The DOF which 



28 



contributes least to the linear independence (i.e., lowest Ed value) is removed from the FEM 
target mode matrix. The Fisher information matrix Aq, the G and Fg matrices, and effective 
independence vector Ed are recalculated and the next sensor location is deleted from the 
target set. This iterative process is performed until the desired number of sensors remains. 
The minimum number of sensors required for identification corresponds to the number of 
target modes supplied. 

The previously described technique chooses single DOFs to place sensors. Often times 
a modal tester uses triaxial sensors instead of single DOF sensors. Assume the FEM being 
used to place sensors has 3 DOF per node. The EI algorithm is modified to choose 3 DOFs at 
a time (corresponding to a node point) which contributed least to the linear independence of 
the target modes were eliminated over each iteration. The EI value for each node is calculated 
as a sum of the EI of each DOF of that node. The Ed values for the 3 DOFs at each node are 
summed as. 



^Dtriax ~ 



Ed(1) + Ed(2) + Ed(3) 
Ed(4) + Ed(5) + Ed(6) 

Ed(s - 2) + Ed(s - 1) + Ed(s) 



(4.2.10) 



The 3 DOFs which contributes least to the linear independence (i.e., lowest EDtnax value) 
are removed from the FEM target mode matrix. However, if 1 of the DOFs for a particular 
node had an EI value of 1 .0, meaning that that DOF was essential to the linear independence 
of the target modes, that node point would be retained, regardless of the ranking of its node 
point EI sum rating compared to the other node points. The Fisher information matrix Aq, 
the G and Eg matrices, and effective independence vector Eotriax are recalculated and the 
next triax sensor location is deleted from the target set. This iterative process is performed 
until the desired number of sensors remains. 

It is suggested that to increase computational efficiency for large FEM, the original 
FEM should be reduced down to a candidate set of measurement locations larger than the 
number of sensors to be placed before performing the effective independence calculations. 



29 



One suggested technique for this reduction is modal kinetic energy, which is discussed in the 
next section. 

4.3 Kinetic Energy 

The use of kinetic energy for optimal sensor placement as well as target mode 
identification has been discussed in several papers (Salamaetal., 1987;Kammer, 1991). The 
modal kinetic energy is calculated using the FEM mass matrix and target modes. The kinetic 
energy of the i'^ DOF of the j^^ mode is given as 

nDOF 



KEij = Oij X ^Ai (4.3.1) 



k=l 
where Oy is the ij"^ entry of the FEM modal matrix O, Mjk is the ik^^ entry of the FEM mass 
matrix M, and nDOF is the total number of DOFs of the mass and modal matrices. The 
kinetic energy matrix, KE, can be expressed as the matrix product 



KE = O (2) MO = 



^^DOFl 
^*^pOF2 

^^DOFn 



(4.3.2) 



where denotes an element by element multiplication of the matrix O and the matrix 
resulting from the product of M and O. The rows of the KE matrix correspond to the DOFs 
of the model and the columns correspond to the modes of the FEM. Locations for actuation 
or sensing are chosen as those DOFs with a maximum value of kinetic energy for a given 
mode. For example, assume the O contains FEM modes 1 through 10 for a given structure 
and one wishes to sense or excite the third mode. The DOF (row) with a maximum kinetic 
energy value for the third mode or column of the KE matrix would be selected. It is assumed 
that by placing the sensors at points of maximum kinetic energy, the sensors will have the 
maximum observability of the structural parameters of interest. 

If the modal test designer wishes to place triaxially constrained sensors, then a KE 
matrix may be calculated by summing the rows of Eq. (4.3.2) corresponding to DOFs for 



30 



each node. Then the node points with maximum KE over the modes of interest may be 
chosen as locations for actuators or sensors. 



^^triax - 



KEpQp, + KEpQp2 + KEqof3 

^^DOFn-2 + KE[3QPn_j + KE^Qp^ 



(4.3.3) 



The kinetic energy objective function precludes placing any sensors or actuators at 
nodal points since there is no motion and zero kinetic energy at these points (i.e., the O entry 
would be zero resulting in a zero product). This could be a limiting factor in the pretest 
planning. To combat this problem, sensors can also be placed using maximum average 
kinetic energy (AKE) technique. A sensor is placed at a DOF with a maximum average 
kinetic energy over a range of modes of interest. In using an average kinetic energy, a DOF is 
not necessarily excluded if it is a node point of a particular mode. The average kinetic energy 
vector is calculated as 



2ke 

k = l 
N 



AKE = -l 



k = l 



Ike 

k = l 



(ndof)k 



/N 



(4.3.4) 



where N is the number of modes in the mode shape matrix O (i.e., the number of columns 
of the KE matrix). The sensor or actuator locations are found by finding the DOFs of the 
maximum average kinetic energies. Triaxially constrained sensors may be placed by taking 
the sum of the average kinetic energy for the DOFs for each node and choosing the nodes 
with maximum average kinetic energy. 

In addition, it should be noted that the mass weighting inherent to the kinetic energy and 
average kinetic energy approaches causes the sensor or excitation placement to become 
dependent on the finite element discretization of the structure. There is an inherent bias 



31 



against the placement of sensors in the areas of the structure in which a fine mesh size (and 
thus small mass) is used. 

4.4 Eigenvector Product 

This technique uses modal products from the reduced FEM eigenvectors to identify 
possible locations for sensors or excitation. By choosing a frequency range of interest and 
the corresponding FEM eigenvectors (or modes) in that range, the eigenvector product is 
calculated as 



EVP = <^ 



^i^(^2<S) . . . ®4>N^ (4.4.1) 



w^here (g) represents an element by element multiplication of the mode shape vectors (j). The 
i^"^ entry of the EVP is given as 

EVPi = (t>ii(|>i24>i3 • • -^^iN (4.4.2) 

This product is calculated for all candidate DOF sensor or actuator locations. A 
maximum value of this product corresponds to a candidate location of reference or excitation 
(Jarvis, 1991). This technique also precludes the placement of sensors at nodal points which 
result in zero eigenvector products. If this presents a problem for a given test case, the 
eigenvector product can be replaced equivalently by an absolute value eigenvector sum, over 
the FE target modes of interest. 

The eigenvector product may be used to place triaxially constrained sensors by 
sunmiing up the entries of Eq. (4.4.2) which correspond to a particular node point. The node 
points with the maximum eigenvector product sum are then chosen as points of reference. 

4.5 Driving Point Residue 

A FEM can be used to identify the best locations and directions for exciting a structure 
by an evaluation of driving point residues (DPRs) or modal participation factors (Kientzy et 



32 



al, 1 989) . A DPR is a measure of how much a particular mode is excited at a particular DOR 
The point and direction of excitation are chosen where the DPRs are maximized (to excite a 
given mode) or minimized (to avoid exciting a given mode). An equation of motion in 
Laplace domain for a structure may be written as 

Ms2 + Cs + k]X(s) = F(s) or B(s)X(s) = F(s) (4.5.1) 

where M, C, and K are the (nxn) mass, damping and stiffness matrices, s is the complex 
Laplace variable (s = a + i(o), and F(s) is the transformed excitation forces. Equation (4.5. 1) 
may be solved for the transformed displacement responses, X(s), 

X(s) = H(s)F(s) where H(s) = B(s)-i (4.5.2) 

and H(s) is referred to as the transfer matrix. The system transfer matrix for a structure with 
damping can be expressed in the form 

^ fRkl fRkl 

H(s)=I^^ + ^ (4.5.3) 

-■ ; ."V - ^ 

where R^ and R^* are the modal residues and X\^ and X[* are the complex conjugate pairs of 
eigenvalues of the transfer matrix. The residues can be written in terms of the mode shapes 
(t)kas 

HM - y ^MM"^ I Kimf (454) 

k= 1 ' - ^k ^-K 

where A^ is the mode shape scaling constant. For a structure which is lightly damped, the 
following two inequalities are true: 

Oj^ <^ 00^ and Imaginaryjcj),^) < Realjcj)^) for k = 1 to N (4.5.5) 

When these conditions are imposed, the mode shape scaling constants can be written in the 
form 

Ak = m^ fork = ltoN (4.5.6) 



33 



and the residues become 

R^(a,b) = [%^^^ fork=ltoN (4.5.7) 

where Rk(a,b) is the residue between DOF a and DOF b, (|)k(a) is the k'*^ mode shape 
component at DOF a, (t)k(b) is the k^^ mode shape component at DOF b. If the mode shapes 
are scaled such that they are mass orthonormal, (i.e., O^M4)=I, where is the matrix whose 
columns are the mode shapes (t)k (for k=l to N), M is the FEM mass matrix, and I is the 
identity matrix) then the residues (in terms of the displacements) may be written as 

Ri^(a, a) = M^ for k= 1 to N (4.5.8) 

or equivalently in terms of acceleration 

Rj^(a, a) = (t)k(a)^tOk for k=l to N (4.5.9) 

The easiest way to evaluate several residues at once is to display them graphically. The 
DPRs that were calculated for the NASA 8-bay truss are shown in Figure 4. 1. The DPRs are 
graphed in order of weighted average residue in order to discriminate against zero DPRs. 
The weighted average residue is calculated as 

waDPR = average DPR x minimum DPR (4.5.10) 

Each vertical line on the graph represents the range of DPRs from maximum to 
minimum over all the modes of interest for a single candidate DOF. The highest weighted 
average which is the best driving point is displayed first. The residues in the top graph are the 
square root of the sum of the squares of the residues in the x, y, and z direction plotted on a log 
scale. The bottom graphs are the residues for the x, y, and z direction plotted on unit 
normalized linear scales. The top graph is used to choose the node at which to place the 
excitation device. The bottom graphs are used to find the x, y, or z direction of the excitation. 
In order to insure that an excitation location will give uniform participation of as many 
target modes as possible, it is desired to find a high average residue for a given DOF as well as 



34 



Weighted Average dprs 



m 6 
Q 

^4 
5, 

TO 2 

_o 


-2 



MH 



({) (j) (j) A (t i cb i " " " " 



C) () () () 



O C) (> () 



() () C) C) 



() () () () 



2 3 4 1 5 8 6 7 13161514191820171110129 21242322252827262932|3l|30 
node 



Q. 

"°2^ 

<D 
N 

lo 

E 
o 



10 15 20 

node 

X,Y,Z dof dprs 



10 



15 20 

node 



25 



25 



30 



z 

- I I I I I I I I 11 I I I I I 11 I I 11 11 I I I I I I I I I 

y 

- I I I I I I I I I I I I III! . 

X 



30 



Figure 4.1 Typical Driving Point Residue (NASA 8-Bay Truss) 



a small residue range over all the modes of interest. For this example the highest weighted 
average DPRs are at nodes 2, 3, 4, and 1 as seen in the top portion of Figure 4. 1 . The bottom 
portion of this figure shows that the optimal directions for excitation at these node points 
would be in the x and/or z direction, because the larger residues are in these directions. 



CHAPTER 5 

MODAL TEST EXCITATION AND SENSOR PLACEMENT: 

NEW TECHNIQUES 



5.1 Introduction 

In an effort to improve on the existing sensor and excitation placement techniques, two 
new sensor placement techniques and two new excitation techniques are developed in the 
current work. The first excitation and sensor placement techniques are based on the FEM 
normal Mode Indicator Function (MIF) calculation. The second excitation and sensor 
placement techniques are based on the observability and controllability calculations of the 
modes of the FEM. The effectiveness of these techniques, along with the techniques 
discussed in Chapter 4, will be explored in subsequent chapters using several different 
structural test-beds. 

5.2 Mode Indicator Function 

The mode indicator function (MIF) was first developed to detect the presence of real 
normal modes in sine dwell modal testing (Hunt et al, 1994 and Williams et al, 1985). This 
function also serves as a useful metric for pre-test analysis. While it is somewhat useful for 
assessing the efficacy of sensor layout, its true utility lies in assessing the effectiveness of a 
particular input in exciting the system modes. 

The first step in calculating the MIF is the calculation of an acceleration frequency 
response function using the FEM mode shapes and frequencies, 

^ - C02(hr{))r 



35 



36 



where m - number of modes in frequency range of interest 

(j)"" - r"^ mode 

^k"^- force input point k of the r^'' mode 

(j)!"" - response point i of the r* mode 

CO - discrete frequency at which to calculate Hjk 

(Or - frequency of the r* mode 

Sr - viscous damping ratio of r'^ mode 

msr - modal mass of the r''^ mode 
Next, the normal MIF is calculated using Hjk as 



i(|Hik(co)2|) 



MIF(a)) = — (5.2.2) 



i = l 

where L is the total number of response points. The MIF is nearly 1 .0 except near a normal 
mode, at which point it drops off considerably since the frequency response becomes mostly 
imaginary at that point (i.e., Real(Hik(co)) is very small). A plot of a typical MIF is given 
in Figure 5.1. 

5.2.1 Excitation Placement 

In pre-test planning, an excitation is desired which exhibits a sharp drop in the MIF at 
each mode of interest, indicating that the mode is well excited. The Genetic Mode Indicator 
Function (GMIF) excitation selection algorithm uses a genetic algorithm (Holland, 1975) to 
find excitation locations and their orientations on a structure to optimally excite a given 
mode or range of modes. The success of the excitation is based on the MIFs of the chosen 
excitation locations. If more than one excitation is sought then a MIF must be calculated for 
each. A single excitation need not exhibit a sharp MIF drop for all modes as long as the union 
of the MIFs for all of the excitation sources exhibits a large drop for each target mode. Two 
algorithms have been developed. The first is an unconstrained version which searches for 



37 




20 30 40 50 

frequency (Hz) 
Figure 5. 1 Typical MIF Plot 



70 



node point excitation locations with forces being applied in any direction at the node points. 
The second algorithm is a constrained version in that the direction of the excitation is 
constrained to be 0, 30, 45, 60, or 90 degrees in each x, y, and z plane. The constrained 
algorithm was developed to provide an improvement in algorithm speed by reducing the 
number of search points in the design space. In addition, the attachment of the excitation 
hardware on the structure during the modal test would be easier if the angles of orientation 
are limited. An outline of the GMIF algorithms follows. 

Coding . The GA chooses an initial population of node points and directions for 
excitation location. The node points and the directions are referred to as design variables. 
The design variables are represented differently for the constrained and unconstrained 
versions of the GMEF algorithm. In both the constrained and unconstrained cases the node 
point locations of the excitations are treated as discrete design variables. Discrete design 
variables represent a finite number of variables to search over, and for this application they 
represent all of the node points in a FEM that are being considered as possible excitation 
locations. The direction design variables are two angles in spherical coordinates, a and p, 
which are used to calculate the direction of the force as seen in Figure 5.2. For the 



38 



F force 




X = r cosa sinp 
y = r sina sinp 
z = r cosp 

F = cosasinPi + sinasinPj. + cosfik 



Figure 5.2 Excitation Selection by GMIF 

constrained algorithm the orientation of the excitation is considered discrete in the sense that 
there are a finite number of angles (i.e., 0, 30, 45, 60, or 90 degrees) from which the GA 
selects a and p. For the unconstrained case, the angles of orientation are considered to be 
continuous in that the GA searches over all possible angles. For both cases the force is 
assigned a unit magnitude in order to only evaluate the angle of orientation of the force and 
not the magnitude. Table 5.1 presents a list of variables used in the GMIF selection 
algorithm. 



Table 5.1 GMIF Design Variable Description 



Unconstrained 


Constrained 


Design Variable 


Type 


Design Variable 


Type 


node 


discrete 


node 


discrete 


a (any angle) 


continuous 


a (0,30,45,60,90 degrees) 


discrete 


p (any angle) 


continuous 


p (0,30,45,60,90 degrees) 


discrete 



Evaluation . The next step in the GA is to evaluate the fitness of each population member 
or excitation. The fitness of a member is based on the calculation of the MEF corresponding 
to each force that makes up a single member. All of the MIFs for a single member are 
assembled into a MIF matrix. 



39 



MIFn,= 



Force 1 

Force 2 (52.3) 

Force nf 



miffi(a)i) miffi((02) • • • miff^(a)ri^ 
mif£2(o)j) m\i^{oi2) ■ ■ ■ mifj2(cori^ 

miff^a)i)miff^co2) . . . miffjjf(u^ 

• ! t t 

natural frequencies of interest 
Next the minimum of each column of MIFm is taken to find the maximum drop-off values 
of the union of the MIFs of each force resulting in a minimum MIF vector, 

MIFv = mTrJiZmCMIF.) (5.2.4) 

The objective function is calculated as a weighted sum of the elements of MIF y. 

m 



Jobj = IwjMIF,. (5.2.5) 



i = l 
The weights may be used to emphasize the drop-off values of particular modes. The 

objective function of Eq. (5.2.5) is designed to find excitation sources which exhibit sharp 

MIF drop-offs for as many modes as possible. 

Selection, crossover, and mutation . Once the fitness of the initial population is 
established the population is allowed to evolve over a fixed number of generations. The 
information contained in the initial population is crossed over between members and sent to 
the next generations. Members of new generations which are more fit than the previous 
generation (i.e., have better drop-off values) replace the less fit members in the evolving 
population. Mutations that occur in the population allow for the population to remain diverse 
during the evolutionary process, keeping the design search space open. 

5.2.2 Sensor Placement 

Once an excitation source has been selected, the MIF corresponding to the chosen 
excitation source may be used along with a GA to locate a sensor set. First, the FRF matrix is 
calculated for the FEM under consideration using the chosen excitations. When the MIF is 
calculated to evaluate an excitation source, all DOFs of the mode shape matrix are used to 



40 



calculate the frequency response matrix, H. When the MEF is used to evaluate a sensor 
placement, only the sensor candidate DOF or three DOFs in case of a triax sensor set, is used 
to calculate the frequency response matrix, H. A MIF must be calculated for each force for a 
candidate sensor and the minimum MIF value for each mode is taken. The MIF values for the 
target modes for the i'*' DOF are taken as the minimum MIF values for all of the forces in an 
excitation set, 



ivfTF — column 
i^iiC-i ~ minimum 



- MIFi(forcei) 

- MIFjCforccj) 

- MIFi(^rce„f) 



(5.2.6) 



The MIF vector of Eq. (5.2.6) is calculated for all candidate sensor DOFs. A weighted sum 
of the MIF values for each DOF is made and assembled into the MIF vector. 



MIFv = < 



^wMIF, 
^ WMIF2 

^ wMIFn 



(5.2.7) 



where w is an (Ixm) weight vector used to emphasize MIF drop-off values. The variable 
n is the total number of candidate sensor DOFs for unconstrained sensor placement or the 
total number of candidate sensor nodes for triaxially-constrained sensor placement. Once 
MIFy has been calculated, the node or dof with minimum MIFy sum is retained as the first 
sensor. The MIFy vector is recalculated using all remaining DOFs plus the single sensor 
chosen, and the next dof or node is chosen with minimum MIFy value. This iterative process 
is performed until the desired number of sensors is chosen. 



5.3 Observability and Controllability 
Consider the set of discrete linear second-order differential equations of motion 



corresponding to a particular «DOF FEM of a structure, 



Mx(t) + Dx(t) + Kx(t) = Bu(t) 



(5.3.1) 



41 



y(t) = Cx(t) 



(5.3.2) 



where M, D, and K are the (nxn) mass, damping, and stiffness matrices, x(t) is the (nxl) 
displacement vector, and u(t) is the («xl) input function of the system. The over dots 
represent differentiation with respect to time. By choosing 



z(t) = 



x(t) 
x(t) 



(5.3.3) 



Eq. (5.3.1) can be rewritten in state space form as 



z(t) = 



I 



z(t) + 




M-^B 



u(t) 



(5.3.4) 



or equivalently 



z(t) = Az(t) + Bu(t) 



(5.3.5) 



where A is the (2nx2«) state matrix, B is the (Inxo) input influence matrix, and u(t) is the 
(oxl) input function vector. The output of the system defined by Eq. (5.3.2) may be rewritten 



as 



y(t) = Cz(t) where C = [C 0] 



(5.3.6) 



A 

The vector y(t) is the (/x7) system output , and C is the (/x2«) output influence matrix. Figure 
5.3 is a pictorial representation of the matrices and vectors of a state space system of 
equations and describes the purpose of each. 

An important consideration in the control of the system described by Eq. (5.3.4) is if the 
system is controllable and observable. Another consideration is the observability and 
controllability of the modes of the system defined by Eq. (5.3.4). Several techniques for 
calculating the observability and controllability of modes have been explored. One of the 
most common tests for controllability and observability is the Popov-Belevitch-Hautus 
(PBH) test (Kailath, 1980). For the purpose of vibration control, it is most common to 



42 




how and where energy 
is injected into system 





how system transforms 
and dissipates energy 



what information is 
extracted from system 



Figure 5.3 State Space Variable Description 

overstep the conversion of Eq. (5.3.1) into state space form and to calculate the observability 
and controllability directly from Eq. (5.3. 1). The PBH eigenvector test for a second-order 
system (Laub and Arnold, 1984) states that given the system defined in Eqs. (5.3.1) and 
(5.3.2): 

1. The i^*^ mode will not be controllable from the j^'^ input if and only if there 
exist a left eigenvector gi such that 



q.'[XfM + XjD + K] = 0' 



q^ = o- 



(5.3.7) 



(5.3.8) 



2. The i^*^ mode will not be observable from the k*^ output if and only if there 
exist a right eigenvector pj such that 



[kfu + Xp + K]p. = 



(5.3.9) 



.T„ _ 



c^Ei = 



(5.3.10) 



where bj is the j"^ column vector of the input influence matrix, B, and c^ is the k'^ column 
vector of the output influence matrix, C. 



43 



This evaluation of controllability and observability tells whether or not the modes are 
completely observable or controllable; it does not address the issue of degree of observability 
and controllability. The issue of degree of controllability and observability is explored in a 
paper by Hamdan and Nayfeh (1989). In this work the matrices Q^B and CP are used to 
evaluate the degree of controllability and observability of the modes of a system. The matrix 
Q-^ is the transpose of the matrix whose columns are the m left eigenvectors of Eq. (5.3.7) and 
B is the output influence matrix whose columns are the o output influence vectors. 



Tia _ 



Q'B 



T 
T 

qm 



I I 
b, bj 

I I 



bo 

I 



(5.3.11) 



The matrix C is the output influence matrix whose / rows contain the output influence vectors 
and P is the matrix whose columns are the m right eigenvectors of Eq. (5.3.9). 



CP = 



-C] - 

- C2 - 

- c, - 



I I 

Pi P2 
I I 



I 

Pm 

I 



(5.3.12) 



The (mxo) matrix Q^B contains information about the controllability of the modes and 
the (Ixm) matrix CP contains information about the observability of the modes. If the ij^^ 
entry of Q^B is then the i''' mode is uncontrollable from the j"^ input. Similarly, if the ki^*^ 
entry of CP is then the i"^ mode is unobservable from the k^'' output. If the ij^"^ entry of the 
controllability matrix is nonzero, then what information may be gained about the degree of 
controllability of the i* mode from the j'*^ input? The ij* element of the Q^B matrix is the 
vector dot product of gj and bj. If the two sub-spaces spanned by these vectors are parallel 
then the i^*^ mode is completely controllable from the j'^ input, and if the two sub-spaces are 
orthogonal then the i^*^ mode is completely uncontrollable from the j^*^ input. If the two 
sub-spaces are neither orthogonal or parallel then the angle between the two is an indication 
of the degree of controllability of the i^^ mode from the j'^ input. This relationship is 
illustrated in Figure 5.4 and the magnitude of the vector dot product is. 



44 



iq;'bji=||q.||||bj||cosei 



(5.3.13) 



A similar argument may be made for the observability of the i^'^ mode from the k'^ output 
using the magnitude of the vector dot product, 



ICkEil =l|Cklli|pJ|cos(j)ki 



(5.3.14) 



The angle Bjj is a direct measure of the degree of controllability of the i"^ mode and ^\^ is a 
direct measure of the degree of observability of i^'^ mode. The degree of controllability and 
observability decrease as Gy and (t)ki go from to n/2 as shown in Figure 5.4. 



CONTROLLABILITY 



OBSERVABILITY 



bj gi 



Ck Ei 



-►-► 



completely controllable 



ey = o 



(l)ki = 



-►-> 



completely observable 









^ 

\ 


4 


^^ 




-^ 




^/■^ 


- ' 






completel) 


u 


nc 


ontrollable 



ey = Till 



(l)ki = 7U/2 




^ 

k 


Ci 


^^ 






^ .."^ 


^y^ 








completely 


u 


[IC 


ontrollable 



Figure 5.4 Controllability and Observability 



The above argument has been made from a dynamic controls perspective. The same 
argument may be used to gain information about actuator and sensor placement during 
modal tests of a structure. Using the FEM of a particular structure the degree of 



45 



controllability of a modal test excitation layout may be used to optimally select an excitation 
location. Similarly, the degree of observability of a modal test sensor layout may be used to 
select a sensor configuration which will result in an increase in the amount of modal 
information obtained. 

5.3.1 Excitation Placement 

The degree of controllability based on the calculation of the angle between the sub-space 
spanned by a mode shape of the system and the sub-space spanned by the input influence 
vectors of the matrix of Eq. (5.3.1) is used to evaluate how effective the input u(t) may be in 
controlling the modes of the system. Consider that Eq . (5 . 3 . 1 ) is the equation of motion for a 
particular structure and that the right hand side, Bu(t), is the force that will be applied to 
excite the structure for modal testing. In order to gain the most information from the modal 
test, an excitation location which will excite a chosen range of target modes well is required. 
The measure of modal controllability is an indication of how great an effect a particular 
input, bj, may have on the mode shapes of the system. An input with a higher degree of 
controllability over a mode will be more successful in exciting that mode than an input with a 
lower degree of controllability over that mode. Therefore, it is proposed that the angle Gy of 
Eq. (5.3.13) may be used as a measure of how successful the input excitation bj will be in 
exciting mode qj. Since there are an infinite number of possible input influence vector 
values, an optimization technique is needed to search for an input influence vector which 
maximizes the controllability of the target modes. A genetic algorithm is employed as the 
optimization tool for this purpose. 

Coding . The coding of the Genetic Controllability (GCON) Algorithm is identical to the 
coding of the GMIF algorithm. One design variable represents the node point locations of 
the forces, the other design variables represent the angle orientations of the forces in 
spherical coordinates as described in Figure 5.2. The difference between the GMIF and the 
GCON algorithms is in the fitness evaluation of the population. 



46 



Evaluation . The fitness of each population member is based on the calculation of the 
controllability vector. The location and orientations of each force in a population member is 
used to calculate an input influence vector. The j * force of a member is used to calculate a 
portion of the input influence vector as. 



b« = 



cosagsinPfj' 

sinttfj sin(3fj 

cosPq 



(5.3.15) 



The unit magnitude vector bg is calculated for all j forces of a member and assembled into 
the global input influence vector. The global input influence vector, b, is initially an n DOF 
vector of zeros. Once the individual force unit input influences are calculated, they are 
placed in the global input influence vector, b, at the DOFs of each corresponding force node 
point, 


bfi 







bfnf 





(5.3.16) 



where nf is the total number of excitation devices represented in a population member. Since 
the magnitude of the bg's affect the controllability of the system, each bfj is scaled to unit 
magnitude so as to compare the directions of the forces as apposed to the magnitudes. The 
unit input influence vector is used in conjunction with the left-hand eigenvectors of Eq. 
(5.3.7) to calculate the (mxl) degree of controllability vector from Equation 5.3.13, 



e = 






(5.3.17) 



47 



The i^*' entry of the degree of controllabiUty vector is 

e. = cos-iSi^ (5-3.18) 

II Hi II 

The entries of the (mxl) controllabihty vector, 0, represents the controllability of each of 
the m modes of the system from the locations and directions defined by b. The algorithm 
is designed to find excitation location which exhibit the highest degree of controllability for 
the modes of interest. Therefore, the most fit members of a population are forces which 
minimize the entries of the vector 9. Doing so minimizes the angle between the input vector 
sub-space and the mode shape sub-spaces thus increasing the amount of controllability and 
the amount of power input into the modes. The objective function is calculated as a weighted 
sum of the entries of 9, 

m 



Jobj = X^i^i (5.3.19) 



i=l 

The weight may be used to emphasize the controllability of particular modes over other 
modes. 

Selection, cross-over, and mutation . The population is allowed to evolve over a fixed 
number of generations as in the GMIF algorithm. The most fit members are those that 
minimize the objective function of Eq. (5.3.19). 

5.3.2 Sensor Placement 

The degree of observability of the modes of the system in Eq. (5.3.1) is based on the 
calculation of the angle between the modes of the system and the output influence matrix. 
When performing a modal test of structure, it is not likely that all DOFs in the FEM will be 
instrumented during the test due to cost constraints. In order to get the most information 
about the modes of the system, a reduced sensor set which has the greatest degree of target 
mode observability should be chosen. Therefore, the angle (^ of Eq. (5.3. 14) will be used as a 



48 



measure of how successful a sensor configuration is in measuring a group of chosen target 
modes. 

There are a finite number of DOFs or sensor possibilities represented in a FEM, 
therefore, an optimization technique is not needed. In order to evaluate each DOF location 
individually, the output influence matrix, C, is set equal to an (nxn) identity matrix. 
Therefore, the observability of the k*^^ DOF of the i^** mode is obtained from Equation 5.3. 15 



as 



*ki = 



llpjl 



(5.3.20) 



where Pki is the ki*^ entry of the right eigenvector matrix of Eq. (5.3.12) and gj is the i"^ 
column of the right eigenvector matrix, P. If this value is calculated for all candidate DOFs 
and all target modes the resulting (nxm) observability matrix, 



<D = 



<^2] <1>22- 



4>nl 4> 



n2- 



• *lm' 

• (l>2m 



. 4)r 



(5.3.21) 



The rows of the observability matrix represent DOFs and the columns represent the 
modes. Once the observability matrix has been calculated, the DOFs for sensor location 
must be evaluated. The observability column corresponding to each mode is sorted, and the 
DOFs with the minimum ^ values (i.e., greatest observability) for each mode are selected as 
sensor locations. 



CHAPTER 6 

PRE-MODAL TEST PLANNING ALGORITHM APPLICATION: 

NASA EIGHT-BAY TRUSS 



6.1 Introduction 

The NASA 8-bay truss is used to compare the techniques discussed in Chapters 4 and 5 
in placing sensors and actuators for modal testing and system identification purposes. The 
kinetic energy, average kinetic energy, eigenvector product, driving point residue, and 
controllability techniques are used to place three excitation devices on the truss. A numerical 
simulation is performed to evaluate the effectiveness of each technique to excite the first five 
target modes of the structure. A cross-orthogonality check between the identified and finite 
element modes is performed in addition to a frequency match comparison. Effective 
independence, kinetic energy, average kinetic energy, eigenvector product, and 
observability techniques are used to place sensors on the 8-bay truss, in order to best identify 
the first five modes of vibration. The structural response of the truss is numerically simulated 
and measured at those DOFs corresponding to the sensor locations obtained using the various 
techniques. The eigensystem realization algorithm (ERA) is then used to evaluate the 
effectiveness of each sensor set with respect to modal parameter identification (Juang and 
Pappa 1 985). A set of three hundred random sensor locations are compared to the five sensor 
location techniques. The cost effectiveness of each of the excitation and sensor selection 
techniques is evaluated. 

6.2 NASA Eight-Bay Test Bed 

The NASA eight-bay truss, pictured in Figure 6.1, is modeled with 96 DOFs and is 
considered to be lightly damped. Using the FE mass and stiffness matrices supplied by 



49 



50 



NASA, the FE mode shapes and frequencies were calculated. When the true modal tests 
were performed on the truss, it was assumed that the first five modes were successfully 
identified (Kashangaki, 1992). Table 6.1 list the first five frequencies and mode 
descriptions. 




Figure 6. 1 NASA 8-Bay Truss 



Table 6. 1 Eight-Bay Truss Frequencies and Mode Description 



Mode 


Frequency (Hz) 


Description 


1 


13.925 


l^^y-x bending 


2 


14.441 


1^'y-z bending 


3 


46.745 


F' torsional 


4 


66.007 


2"'^ y-x bending 


5 


71.142 


2"'! y-z bending 



6.2.1 Excitation Placement 

Kinetic energy, average kinetic energy, eigenvector product, driving point residue, and 
controllability techniques are each used to place three excitation devices on the 8-bay truss to 
best excite the first five modes of the structure. The excitation locations for the five 
techniques are pictured in Figure 6.2. The kinetic energy technique placed two excitation 



51 



Kinetic 
Energy 



Average Kinetic Energy 

and 

Controllability 



Eigenvector 
Product 



Driving Point 
Residue 




Figure 6.2 Eight-Bay Excitation Locations 



sources towards the cantilevered end of the truss and one towards the center of the truss. The 
other four techniques clustered all of the excitation sources at the cantilevered end of the 
truss. It is interesting to note that the kinetic energy technique put all excitation sources in the 
z-direction. The remaining four techniques placed excitation sources in both the x and z 
directions, in addition to clustering two of the sources at a single node. The average kinetic 
energy and controllability techniques placed the excitation devices in the same location as 
seen in Figure 6.2. In a true modal test, the two excitation sources which were placed at a 
single node could be combined into one excitation source at that particular node in the 
xz-direction, thus reducing the number of excitation sources needed. The kinetic energy 
placement could not result in this option. ' 



52 



Once the excitation sources have been determined using the five techniques, the truss's 
response to an impulse at the chosen excitation locations is numerically simulated for the first 
five modes and 5% noise is added to the response data. The response, measured at all 96 
DOFs to fully evaluate the effectiveness of the each excitation placement, is sampled for a 
length of 2 seconds at 200 Hz. Five percent noise is added to the response which is sent to 
ERA for system identification. A comparison of the measured frequencies and 
cross-orthogonalities of the identified and FE models is calculated for each excitation 
placement. 

The five excitation placement techniques were all successful in exciting the structure at 
the target frequencies based on the comparison of the original FE and identified frequencies 
and mode shape. All of the techniques had excellent matching between identified and FEM 
frequencies with differences much lower than the industry standard of 5% (Flanigan and 
Hunt, 1993). A Frequency Response Function (FRF) is plotted for each of the excitation 
devices in Figure 6.3. The cross-orthogonalities between identified and FEM mode shapes 
for the five excitation placement techniques are pictured in Figure 6.4. All off-diagonal 
terms for each of the excitation sets are less than or equal to 0.02, which is well within the 
industry standard of <0.02 off-diagonals for primary modes (Flanigan and Hunt, 1993). 

Of the five techniques, the cross-orthogonality of kinetic energy was worst although it 
was well within the acceptable range of off-diagonal values. The cross-orthogonality of 
average kinetic energy, controllability, eigenvector product, and driving point residue 
techniques had similar values; all of the off-diagonal elements for each of the techniques 
were <0.01. Recall that kinetic energy placed all excitation sources in the z-direction at 
three separate nodes, and that the other three location techniques placed excitation sources in 
both the X and z directions and collocated two sources at one node. The similar placement 
configurations at the cantilevered tip of the truss resulted in similar cross-orthogonalities. 
Based on the frequency matching and cross-orthogonality between identified and FEM 
frequencies and mode shapes, and on the FRFs, each of the five excitation location 



53 



Kinetic Energy 




o 

e 
a 

■4-» 

u 
o 



. Average KE and Controllability 




Frequency(Hz) 
Eigenvec tor Product 



Frequency(Hz) 
Driving Point Residue 




50 60 70 80 



Frequency(Hz) 



Frequency(Hz) 



Figure 6.3 Eight-Bay Excitation Location Frequency Response of Time Domain Data 



Kinetic Enere 




Eigenvector Product 




Mode 



Average Kinetic Energy and Controllability 



1.00 



0.02 
0.00 

a 1.00 
m 0.02 
H 0.01 
■<0.01 



1.00 



0.02 
0.00 




Mode 
Driving Point Residue 




Mode 



Figure 6.4 Eight-Bay Excitation Placement Cross-orthogonality of Identified and 
FEM Modes 1 to 5 



54 



techniques identified an acceptable three point excitation location for exciting the first 5 
modes of the 8-bay truss. 

6.2.2 Sensor Placement 

Using the FE modes and frequencies, the five sensor placement techniques, effective 
independence, kinetic energy, average kinetic energy, eigenvector product, and 
observability were assigned the task of best identifying the first 5 modes of vibration by 
placing 15 sensing devices on the truss. The five sensor set configurations are pictured in 
Figure 6.5. Each of the techniques clustered the sensors in two locations on the truss at the 
cantilevered end and at the midspan. Effective independence, average kinetic energy, and 
eigenvector product techniques collocated fourteen of the fifteen sensors at seven node 
points. The kinetic energy technique collocated twelve of the fifteen sensors at six node 
points and observability collocated eight sensors at four node points. From a cost standpoint, 
the collocation of as many sensors as possible is desired. None of the five techniques placed 
sensors in the y DOE This is to be expected since the first five modes do not include 
significant motion in the y-direction. 

The simulated response, with 5% noise added, obtained using the excitation 
configuration determined from the average kinetic energy technique was used to test the 
sensor sets. The exact same response was used to test each sensor configuration, by taking 
from the 96 DOF response only those DOFs corresponding to the sensor locations to be 
evaluated. The response data of each sensor set were sent to ERA for frequency and mode 
shape identification. 

The excitation placement found using the average kinetic energy technique is used to 
excite the structure to test the various sensor locations. Each of the five sensor placement 
techniques measured the target frequencies well as can be seen from the percent difference in 
the FE and identified frequencies given in Table 6.2. The 96 DOF FEM mass matrix was 
reduced to a 15 DOF mass matrix using exact reduction and the cross-orthogonality between 



55 



i" 



Effective 
Independence 



Kinetic 
Energy 



Average Kinetic 
Energy 



Eigenvector 
Product 



Observability 




xz 



xz 



Figure 6.5 Eight-Bay Sensor Locations 

identified and FEM modes 1 through 5 was calculated using the reduced mass matrix. The 
cross-orthogonality for each of the sensor sets is given in Figure 6.6. 

For each of the five cases the cross-orthogonality terms were within acceptable limits. 
The off-diagonal terms corresponding to the primary modes were all =^ 0.02. Effective 
independence, kinetic energy, and eigenvector product techniques resulted in similar 
cross-orthogonalities (all off-diagonals are =^ 0.01). Observability technique gave the worst 
cross-orthogonality results of the five techniques even though the off-diagonal elements 



56 



remained within acceptable values. The improved performance of the effective 
independence, kinetic energy, observability, and eigenvector product techniques over the 
average kinetic energy technique can clearly be seen in the next section when the five 
techniques are compared to the random sensor sets. 



Effective Independence 



Kinetic Energy 




1.00 



0.02 
0.00 



Mode 




Average Kinetic Energy 




Mode 



1.00 



0.02 
0.00 



Mode 



Eigenvector Product 




Observability 



1.00 



0.02 
0.00 




a 1.00 
B 0.02 
■ 0.01 
■<0.01 



Mode 



Figure 6.6 Eight-Bay Sensor Placement Cross-orthogonality 
of Identified and FEM Modes 1 to 5 



57 



Table 6.2 Percent Difference in FE and Identified Frequencies 



MODE 


EI 


KE 


AKE 


EVP 


CON 


1 


0.042 


0.052 


0.066 


0.042 


0.17 


2 


0.018 


0.004 


0.012 


0.018 


0.19 


3 


0.141 


0.036 


0.013 


0.142 


0.30 


4 


1.140 


0.032 


0.455 


1.140 


0.79 


5 


0.021 


0.008 


0.037 


0.021 


0.15 



6.2.3 Results: Random Sensor Location 



Three hundred random sets of 1 5 sensors each were generated and evaluated in order to 
assess the level of increased performance of the various sensor placement algorithms against 
pure chance. The same time domain response with 5% noise used in the previous section was 
partitioned to the random sensor configurations. For the time domain data of each sensor set, 
ERA is used to identify the first five frequencies and mode shapes. The cross-orthogonalities 
and frequency differences between the identified and FEM modes and frequencies were 
calculated. Figure 6.7 is a comparison of cross-orthogonalities for the 300 random sensor 
sets and for the five sensor placement techniques (EI,KE, AKE, EVP, and OBS). The bar 
portion of each graph corresponds to each random sensor set value and the straight lines 
correspond to the five evaluated sensor configurations (EI, KE, AKE, EVP, and OBS) and to 
the average value of all the random sensor sets (ARS). The top graph of Figure 6.7 is a plot of 
the maximum off-diagonal elements of the cross-orthogonality matrix, the center graph is 
the average off-diagonal of the cross-orthogonality matrix, and the bottom graph is the two 
norm of the cross-orthogonality matrix minus the identity matrix. 

In general, the average random sensor set was within the acceptable limits on frequency 
matching and cross-orthogonality. This is due to the fact that a large number of sensors were 
placed on the truss and approximately 1/3 of the trusses node points would be instrumented 
by the random sensor sets. Statistically, the random sets would have good chances of 
capturing pertinent modal information. Of the five techniques evaluated, kinetic energy. 



58 



effective independence, eigenvector product, and observability gave better results than 97% 
of the random sets as can be seen in Figure 6.7. The maximum off-diagonal and the average 
off-diagonal of the cross-orthogonality matrices of the three techniques are less than those 
for the average random set. In addition the two norm of the difference between the 
cross-orthogonality and the identity matrix for the three techniques is lower than that of the 
average random set. However, the average kinetic energy gave results similar to the average 
random configuration, and showed little to no improvement over the purely random 
placement of fifteen sensors. The maximum off-diagonal of the AKE set was larger than that 
for the average random set and the average off-diagonal and two norm were approximately 
equal to those of the average random set. 

6.3 Computational Efficiency 

For this particular example, the results for each of the excitation and sensor placement 
techniques are all relatively comparable. The targeted modes and frequencies are excited by 
all of the excitation placements and properly identified by all of the sensor sets evaluated. 
This is illustrated by the acceptable differences in FEM and identified frequencies and 
cross-orthogonality values. The agreement between all the techniques can be partially 
contributed to the fact that the "the modal test" was a numerical simulation. The differences 
in the results for excitation evaluation and identification may be greater for a true modal test. 
However, an important issue that must be considered when using the discussed techniques 
for excitation and sensor location is the computational cost of each evaluation versus the 
accuracy of the modal test results. As can be seen from Table 6.3, the most efficient 
technique for excitation and sensor location is the eigenvector product technique. It may 
well be that on a more complicated example, the more computationally efficient techniques 
may result in modal test configurations which give less accurate modal information than the 
more complex placement techniques. A larger system model with more DOFs may make the 



59 



0.07 



Cross-Orthogonality: Maximum Off-Diagonal 




Cross-Orthogonality: Average Off-Diagonal 



0.012 




Two Norm: Cross-Orthogonality - Identity 




ARS 

AKE 

iLUMimiim iyiiioBs 

EVP 



3(X) 



Figure 6.7 Eight-Bay Cross-Orthogonalities of Five Techniques 
Compared to 300 Random Sensor Sets 



60 



tradeoffs between the computational cost of placement and the accuracy of the modal 
identification more apparent. 

Table 6.3 Total Floating Point Calculations for Each Placement Technique 



Technique 


Total flop count 


Placement 


EI 


812,000 


sensors 


KE 


92,700 


sensors & excitation 


AKE 


93,200 


sensors & excitation 


EVP 


480 


sensor & excitation 


DPR 


7,600 


excitation 


CON 


7,600 


excitation 


OBS 


7,600 


sensor 



6.4 Conclusion 

Based on the evaluation of the numerical simulation, each of the five excitation 
techniques successfully placed three excitation sources on the structure which would excite 
the first five modes of vibration. The sensor placement techniques of effective 
independence, kinetic energy, eigenvector product, and observability found sensor locations 
which showed better frequency matching and cross-orthogonality than 97% of the random 
sensor sets. The sensor set obtained using average kinetic energy showed no improvement in 
cross-orthogonality or frequency matching over those of the random sensor sets. Based on 
the similar results of the placement techniques for sensors and actuators, a more complex 
structure will now be used to compare the techniques discussed in this chapter as well as other 
techniques outlined in Chapter 5. 



CHAPTER 7 

PRE-MODAL TEST PLANNING ALGORITHM APPLICATION: 

MICRO-PRECISION INTERFEROMETER TRUSS 



7.1 Introduction 

A comparative study of several pre-modal test planning techniques is presented using 
the Jet Propulsion Laboratories' Micro-Precision Interferometer (MPI) testbed. Mode 
indicator functions calculated using a reduced FEM of the structure and degrees of target 
mode controllability are used in conjunction with genetic algorithms to find location and 
orientation of two excitation sources in order to optimally excite a chosen range of target 
modes during a modal test. Effective independence, kinetic energy, eigenvector product, 
observability, and mode indicator function techniques are used to place a combination of 
sensors on the structure for the purpose of modal identification. The sensors are placed in 
two ways: independent sensor placement and triaxially constrained placement. A numerical 
simulation of the response of the structure is used to evaluate the effectiveness of each of the 
placement techniques to identify the target modal parameters of the structure. The effect of 
FEM error on the various placement techniques is evaluated. 

7.2 Micro-Precision Interferometer Test Bed 

The MPI, shown pictorially in Figure 7. 1 , is a testbed that has been built in order to study 
structural control systems in the development of space interferometers. Modal tests were 
performed on the MPI structure by two independent groups (Sandia National Laboratories 
and the Jet Propulsion Laboratories (Red-Horse et al., 1993; Carne et al., 1993; Levine-West 
etal., 1994). 



61 



62 




right extending boom 



left extending boom 



Figure 7.1 MPI Structure 

The FEM used to evaluate the placement techniques in the current work was obtained 
from Sandia National Laboratories (Red-Horse et al., 1993). The model used is a 240 DOF 
Guyan-reduced FEM which has been updated using the data obtained from the modal test of 
the structure. The 240 DOFs correspond to three DOFs (x,y,z) at each of the 80 node balls. 
The frequencies from the Guyan-reduced FEM corresponding to the first 12 non-rigid-body 
modes are given in Table 7.1 and are compared to actual frequencies obtained during the 
modal test. 

7.3 Excitation Placement 



During the original modal test of the MPI structure, two excitation sources were used as 
pictured in the top portion of Figure 7.2. The lower portion of this figure is the excitation 
configurations that were obtained by optimizing the MIFs of the FEM using a GA (GMIF) 
and by optimizing the modal controllability of the FEM using a GA (GCON). Both the 
original and the GMIF excitation locations have an exciter on the two extending booms 
although they are oriented differently. The GMIF set-up moves the excitation of the right 
extending boom to an interior point in comparison to the original configuration. The GCON 



63 



technique placed an exciter at the mid-point of the left extending boom and an exciter at the 
top of the main boom. Figure 7.3 gives typical frequency responses for the two excitations of 
the three techniques shown in Figure 7.2; the responses are measured at the sensor location 
shown in Figure 7.2 in the y-directions. The straight line corresponds to the first force 
located by each technique and the dotted line corresponds to the second. 



Original m 


od 


al test excitation 




5j 




sensor 








node 41 








0.707 X 


node 79 / 
0.000 X /) 


K;^^ 


\/> 


. 0.000 y 
\_^ 0.707 z 


0.707 V /A4^^ 


^^j^o^^^^^^^^^ 


0.707 z^j^m^ 


V 







GMIF derived excitation 



GCON derived excitation 



node 77 

0.5225 X 
0.8323 y 
0.1538 z 



sensor 




node 19 
0.2471 X 
0.9618 y 
0.1178 z 



sensor 



node 6 

0.8660 X 

-0.5000 y 

0.0000 z 




node 67 

0.000 X 

0.000 y 

-1.000 z 



Figure 7.2 Excitation Placement on MPI Structure 



64 



Table 7. 1 Reduced MPI FEM Frequencies Compared with MPI Modal Test Frequencies 



Mode 


Frequency (Hz) 
FEM 


Frequency (Hz) 
Modal Test 


1 


7.82 


7.75 


2 


11.66 


11.65 


3 


12.75 


12.67 


4 


29.52 


29.36 


5 


34.45 


34.06 


6 


37.76 


37.34 


7 


42.81 


42.25 


8 


47.30 


46.04 


9 


51.14 


50.69 


10 


52.36 


53.00 


11 


55.41 


56.82 


12 


61.40 


60.04 



The excitation devices placed by the GMIF algorithm were selected to minimize an 
objective function which was dependent on the MIF of each of the two excitation locations. 
The MIF will be nearly 1.0 except near normal modes, at which point it drops off 
considerably. This drop-off indicates that the mode is well excited. Therefore, it is desirable 
to find two excitation sources (location and orientation) which exhibit a sharp drop at all of 
the normal frequencies. The GMIF objective function was designed to find an excitation 
source(s) which exhibits sharp drop offs at normal frequencies as discussed in Chapter 5. 
The GCON algorithm, as discussed in Chapter 5, was designed to find excitation sources 
which minimize the angles between the input influence vector subspace and the sub-spaces 
spanned by the modes of the system thus maximizing the controllability of the modes of the 
structure. By choosing excitations with maximum FE modal controllability, the amount of 
energy being imparted to the FE modes of the system by the excitation is theoretically 
maximized. In order to compare the three excitation sources, the MIF drop off values for the 
first twelve modes of each of the excitations is calculated and shown in Table 7.2. The 
sharpest drop-off value for each of the modes is highlighted in bold in the table. 



65 



Original Modal Test Excitation 




•O 10' 



frequency (Hz) 



GMIF Excitation 




frequency (Hz) 
GCON Excitation 




node 6 
node 67 



frequency (Hz) 
Figure 7.3 Typical Frequency Response for MPI Structure 



66 



Table 7.2 Original, GMIF, and GCON Excitation Locations MIF Values 





Original 


GMIF 


GCON 


MODE 


41 


79 


19 


77 


6 


67 


1 


0.0014 


0.0006 


0.0003 


0.0106 


0.0345 


0.0129 


2 


0.0063 


0.2601 


0.0105 


0.0048 


0.0095 


0.0079 


3 


0.0394 


0.0022 


0.4106 


0.0428 


0.0211 


0.0256 


4 


0.0854 


0.2017 


0.0228 


0.0706 


0.4023 


0.0287 


5 


0.0236 


0.0420 


0.0836 


0.0184 


0.0174 


0.0408 


6 


0.4636 


0.0609 


0.0393 


0.6855 


0.2495 


0.0862 


7 


0.0484 


0.0489 


0.0911 


0.7126 


0.6753 


0.0372 


8 


0.0755 


0.3263 


0.3318 


0.0644 


0.0996 


0.1947 


9 


0.1537 


0.7790 


0.7521 


0.0824 


0.0550 


0.5615 


10 


0.7228 


0.8957 


0.1674 


0.8050 


0.7625 


0.2260 


11 


0.4077 


0.0407 


0.0826 


0.2747 


0.4372 


0.0747 


12 


0.0724 


0.8088 


0.4408 


0.0591 


0.0703 


0.2013 


min MIF 


2/12 


7/12 


3/12 



The GMIF excitations exhibit a sharper drop-off than the original excitations' MIFs for 
10 of the 12 target modes. The GCON exhibit a sharper drop-off than the original excitation 
for 6 of the 12 modes. Comparing all three techniques, the GMIF had the most minimum 
drop-off values at 7 followed by the GCON technique at 3, and the original excitation at 2. 
This is not a surprise since the GMIF is designed specifically to find excitation sources which 
exhibit the greatest drop-offs. An improvement in the drop-off of the GMIF excitation over 
the original excitation can especially be seen for the tenth mode. 

It is interesting to note that even though the GMIF has the best overall MIF drop-off 
values, the minimum drop-off values of the GCON excitations are well within acceptable 
levels. The highest minimum value for the GCON technique is 0.25 for the fifth mode. 
However, in the next chapter it will be shown that an excitation with good controllability 
values does not necessarily have acceptable MIF values. In order to evaluate the 
performance of the genetic algorithm for excitation placement, a set of 500 random 2-point 



'i. 



l_ 



67 



excitations is generated. The number of random excitations, 500, was chosen because the 
GMEF algorithm evaluated approximately 400 population members in the search for the 
chosen GMIF excitation. The MIF values for each of the random excitations are calculated, 
sorted and graphed in Figure 7.4 and the controllability angles for the random excitations are 
calculated, sorted, and graphed in Figure 7.5. 

The top graph of Figure 7.4 shows the maximum MIF value for the original, the GMIF 
derived, and the GCON derived excitations superimposed on the graph of 500 random 
designs in order to compare the values. The top portion shows the GMIF excitation has a 
smaller maximum MIF than 97% of the random population. The bottom portion of Figure 
7.4 is a graph of the genetic MIF excitation placement algorithm objective function values of 
the random and selected excitations. An optimal excitation according to the GMIF objective 
function is one which has as small as possible maximum MIF drop off value. The bottom 
portion of the figure shows that the GMIF excitation outperformed 100% of the random 
population. This shows that the genetic algorithm was successful in finding a good 
excitation based on the objective function designed in a more computationally efficient 
manner than an exhaustive search. 

The same evaluation was performed for the controllability angle calculations as pictured 
in Figure 7.5. The controllability excitation placement algorithm objective function is 
designed to find excitations which have a minimum controllability angle sum over the target 
modes to be excited. Even though the original and GMIF excitations have a smaller 
maximum controllability angle as seen in the middle graph of Figure 7.5, the GCON 
excitation has a better angle sum as seen in the bottom graph. In fact, the GCON excitation 
and the original excitations outperform 100% of the randomly generated excitations. 

Numerical simulations of the MPI structural response to simultaneous impulses applied 
at the two excitation locations were calculated within the MATLAB environment. Five 
percent noise was added to the simulated time responses of the structure. These time 
responses were used along with the Eigensystem Realization Algorithm (ERA) to identify 



68 



1 

0.8 
^0.6 
^0.4 

0.2 






Maximum MIF 




1 1 1 r 1 1 1 


~ 


-^^ : 


- ^^--^ 


- 


1 1 1 1 1 1 1 1 1 



2.5 

2 

& 

1 
0.5 



EXCITATION 

random 
original 



GCON 

GMIF 



50 100 150 200 250 300 350 400 450 500 

GMIF Objective Function 






50 100 150 200 250 300 350 400 450 500 

Random Excitations 



random 



original 

GCON 
GMIF 



Figure 7.4 Comparison of Selected Excitation and Random Excitation 
MIF Values 



the twelve target mode shapes and frequencies (Juang and Pappa, 1985). The evaluation of 
the success of ERA to identify the frequencies and mode shapes was based on a frequency 
percent difference comparison between identified and FE frequencies and on a 
cross-orthogonality check between FE and identified mode shapes using an exactly reduced 
mass matrix. The reduction is exact in the sense that the frequencies and mode shapes of the 
reduced system match exactly their counterparts in the unreduced model (O'Callahan et al., 
1989). 

When ERA was used to identify system mode shapes and frequencies, it missed the fifth 
and tenth frequencies and mode shapes when the original 41/79 DOF excitation locations 
were used to numerically simulate structural excitation. To illustrate this, the 
cross-orthogonality between FEM and ERA identified mode shapes was calculated, and is 



69 



Minimum Controllability Angle 




EXCITATION 

random 

GMIF 

original 
GCON 



50 100 150 200 250 300 350 400 450 500 



90 



on 






89 


Lh 




6U 




9) 




3 


88 


U 








Dl) 




C 


8/ 



86 



1080 



gioeo 



T 1 I I I I - r T r ^^^ 



50 100 150 200 250 300 350 400 

GCON Objective Function 

— 1 1 "^ -r- 1 1 — 



450 



100 



150 200 250 300 350 400 

Random Excitations 



500 




450 500 



random 
GCON 



GMIF 

original 



random 
GMIF 



original 
GODN 



Figure 7.5 Comparison of Selected Excitation and Random Excitation 
Controllability Angles 



pictured in Figure 7.6. For this example, the 240 DOF simulated response was partitioned to 
the sensor configuration obtained using the EI technique as discussed in the following 
section. These poor cross-orthogonality results are corroborated by the frequency response 
function shown in Figure 7.3 in which poor excitation can be seen for modes 5 and 10. The 
GMIF and GCON derived excitations resulted in successful ERA identification of all 12 
mode shapes and frequencies of the original FEM. 



70 




modes 



Effective Independence 
Unconstrained Sensor Set 



n 1.00 

11 <0.50 

■ <0.25 

■ <0.02 



Figure 7.6 Cross-Orthogonality Between FE Modes and Identified Modes 



7.4 Sensor Placement 

Six sensor selection techniques, effective independence, kinetic energy, average kinetic 
energy, eigenvector product, observability, and MIF were used to place sensors on the MPI 
structure. Most of these techniques were previously evaluated for sensor placement using 
the NASA eight-bay testbed in Chapter 6. In that study, the five techniques of effective 
independence, kinetic energy, average kinetic energy, eigenvector product, and 
observability, performed equally well. This could be due to two reasons: (i) the structure 
lacked significant dynamic complexity required to distinguish between the methods or (ii) 
the methods were actually so similar that they led to similar results regardless of structural 
dynamics. One purpose of this study is to again evaluate the five techniques on a more 
complex dynamic system in addition to evaluating the efficacy of using the MIF for sensor 
placement. The second purpose is to investigate the suitability of the techniques when the 
sensors are constrained to be placed in a triaxial fashion. 

Eighteen sensors were placed in two different studies using the six techniques in order to 
best identify the 1 2 target FEM mode shapes and frequencies. First, the techniques were used 
to choose 18 of the 240 DOFs as sensor locations. In the second study, the techniques were 



71 



constrained to choose 18 triaxially constrained sensors (i.e., 6 triax-sensor sets). The 
excitations selected using the GMIF discussed in the previous section were used to excite the 
MPI structure numerically in order to test the various sensor configurations. 

7.4.1 Unconstrained Sensor Placement 

The first placement study evaluated the six techniques' placement of 18 sensors on the 
MPI structure at any of the 240 DOFs (x,y,z of the 80 node balls). The first 12 flexible modes 
of vibration v^'ere chosen as the target modes for each technique. The locations of the sensors 
obtained using each of the techniques are pictured in Figure 7.7. 

All of the techniques evaluated placed a majority of the 1 8 sensors at the ends of the three 
booms. In addition all of the techniques except MIF placed sensors in the two DOFs for each 
boom which exhibited the greatest range of motion (i.e., xy for the primary boom, xz for the 
extending right boom and yz for the extending left boom). The EVP technique clustered all 
18 sensors at the boom tips, and the AKE techniques clustered 17 of the 18 sensors at the 
boom tips, with one sensor being placed near the mid-span of an extending boom. The KE 
technique placed 15 sensors at the boom tips with 3 sensors near the mid-span of the two 
extending booms. The EI technique placed 1 3 sensors at the boom tips and at least one sensor 
near the mid-span of the main and extending booms. The observability technique placed 17 
of the 18 sensors at the boom tips and 1 sensor at the mid-span of an extending boom. The 
MIF technique resulted in the most unusual sensor configuration with several sensor being 
placed in the z-direction on the main boom. The z-direction is not the primary direction of 
motion for this portion of the truss. 

Of the twelve target modes shapes, modes 2 through 1 1 exhibit a bending mode similar 
to that of second-mode-cantilevered-beam bending in at least one of the main or extending 
booms. Second-mode bending is clearly exhibited by the two extending booms for all target 
modes and is exhibited by the main boom for some of the twelve target modes. The sensor 
configurations chosen by the EI and KE techniques are particularly suited to capture this 



72 



second-bending mode shape due to their placement of some sensors at mid-spans of the three 
booms. 

The FEM of the MPI structure was used with MATLAB to simulate a time response of 
the structure to an impact applied at the GMIF chosen excitation locations for all 240 DOFs. 
Five percent uncorrected noise was added to the time response. The noisy response was then 
partitioned to each of the six sensor sets and was sent to ERA for mode shape and frequency 
identification. 

All the techniques resulted in percent frequency difference between FEM and identified 
frequencies of much less than 1 % which is well within industry accepted standards (Flanigan 
and Hunt, 1993). Cross-orthogonalities between FEM and identified mode shapes were 
calculated for each of the techniques and are pictured in Figure 7.8. In order to calculate the 
cross-orthogonalities, the 240 DOF FE mass matrix was reduced to 12 DOFs using exact 
reduction. For this size model, exact reduction was computationally acceptable, therefore, it 
was used to get the best cross-orthogonality comparison. 

All of the off-diagonal elements of the cross-orthogonality matrix for the EI technique 
are within the industry accepted standards of <0.02 for primary modes (Flanigan and Hunt, 
1993). This can be seen graphically in Figure 7.8. For the KE, AKE, and MIF techniques, 
almost all of the off-diagonal elements are <0.02. Some of the entries are between 0.02 and 
0. 1 which is within the industry standard for secondary modes (<0. 1). The OBS technique 
resulted in the most modes having cross-orthogonalities <0.10 which is acceptable for 
secondary modes. The 8'^ mode was not successfully identified by the OBS sensor set. The 
cross-orthogonality for the EVP technique was poor for all target modes. The 
cross-orthogonality of the EVP technique was evaluated with no noise, 1%, 2%, and 5% 
noise added to the time response. Of the four time responses evaluated, only the response 
with no noise gave acceptable cross-orthogonality values. Based on these calculation, the 
EVP technique was unsuccessful in finding an acceptable sensor set. 



73 



Effective Independence 



Eigenvector Product 




Kinetic Energy 



Average Kinetic Energy 




Observability z 



MIF 




Figure 7.7 Unconstrained MPI Sensor Sets 



74 



a) Effective Independence 




□ 1.00 

■ <0.10 

■ <0.02 



c) Average Kinetic Energy 



0.10 
0.02 



□ 1.00 

■ <0.10 

■ <0.02 



modes 



Figure 7.8 Cross-Orthogonality Between MPI FE and Identified Modes 
18 Unconstrained Sensors 



75 



H, 



d) Eigenvector Product 



1.00 
<0.75 
<0.50 
<0.25 
<0.02 



e) Observability 



D 



1.00 
<0.50 
<0.25 

<0.10 
<0.02 



e) MIF 



□ 1.00 

■ <0.10 

■ <0.02 




modes 



0.10 
0.02 



Figure 7.8 — continued 



76 



7.4.2 Triaxially Constrained Sensor Placement 

The six sensor placement techniques were modified to place 6 triaxially constrained 
sensor sets (18 total sensors) at any of the 80 node balls of the MPI structure. The resulting 6 
triax-sensor sets placed using the six placement techniques are pictured in Figure 7.9. The 
EI, KE, and AKE techniques grouped two sensor sets at or near the end of each boom, the 
OBS technique put one, two, and three sensors at the end of each boom, and the EVP 
technique placed three triax-sets at the ends of only two of the booms. It should be noted that 
if the EVP placement task were extended to placing 7 triax-sets, the seventh set would be 
placed at the end of the main boom using the EVP technique. The MIF sensor placement 
technique put two sensors at the end of the right boom, one sensor at the end of the left boom, 
one sensor at the base of the left boom, and two sensors on the main boom. As in the case of 
the unconstrained set, the MIF technique did not place any sensors on the tip of the main 
boom. r 

The time response of the MPI structure excited by the GMIF actuator locations was 
partitioned to those DOFs corresponding to the six triax sensor locations chosen by the six 
placement techniques. The partitioned numerical data with noise added was sent to ERA in 
order to evaluate the effectiveness of each of the triax-sensor sets in identifying the system 
mode shapes and frequencies. All the techniques resulted in percent frequency difference 
between FEM and identified frequencies of much less than 1 % which is well within industry 
accepted standards. The cross-orthogonality calculations between the FEM target modes 
and the ERA identified modes were performed using an exactly reduced mass matrix as in the 
previous section, and are shown in Figure 7.10. All of the off-diagonal cross-orthogonality 
values for the KE techniques, shown in Figure 7.10, were within the industry standard of 
<0.02 for off-diagonal elements for primary modes. 

The EI, AKE, and MIF techniques resulted in cross-orthogonalities which were within 
this standard for most of the modes, but which were slightly above the off-diagonal standard 
for a few modes. The OBS technique resulted in acceptable cross-orthogonalities for most 



77 



modes and acceptable secondary cross-orthogonalities for modes 7 through 10. The EVP 
technique resulted in poor off-diagonal cross-orthogonality values for all modes. 



Effective Independence 



Eigenvector Product 




Figure 7.9 Triaxially Constrained MPI Sensor Sets 



78 



a) Effective Independence 



□ 1.00 

■ <0.10 

■ <0.02 



b) Kinetic Energy 




D 1.00 



c) Average Kinetic Energy 



1.00 



D 1.00 



<0.10 
<0.02 



modes 



Figure 7.10 Cross-Orthogonality Between MPI FE Modes and Identified Modes 
6 Triaxially Constrained Sensors 



79 



d) Eigenvector Product 



D 1.00 

□ <0.75 

■ <0.50 

■ <0.25 

■ <0.02 



e) Observability 




D 1.00 

■ <0.10 "^odes 

■ <0.02 

f) Mode Indicator Function 



D 1.00 

■ <0.10 Ixodes 

■ <0.02 



Figure 7.10 — continued 



80 



7.4.3 Unconstrained vs. Triaxially-Constrained Sensor Sets 

Based on the cross-orthogonalities and frequency differences between FE and identified 
mode shapes and frequencies the EI, KE, AKE, and MIF techniques located sensor sets for 
the unconstrained and constrained examples which were successful in identifying the target 
mode set. Only a few of the cross-orthogonalities were slightly above acceptable primary 
mode values, and all values were within secondary mode standards. For the unconstrained 
sensor set, the EI sensor set resulted in identified modes with the best cross-orthogonality 
with the FE target modes. However, for the triaxially-constrained example, the KE sensor 
set resulted in identified modes with the best cross-orthogonality with FE target modes. In 
both constrained and unconstrained cases the EVP technique resulted in identified modes 
with poor cross-orthogonalities with FE mode shapes. However, the EVP and OBS triaxially 
constrained sensor sets showed an improvement over the unconstrained sets. This is unusual 
because as a rule, the constrained sets do not perform as well as the unconstrained sets. 

7.5 Effect of Model Error 

In order to investigate the effect that model error has on the various placement 
techniques, error was added to the original Guyan-reduced FEM of the MPI structure as seen 
in Figure 7.11. Specifically, 1/3 of the struts' cross-sectional areas were decreased by 20%, 
1/3 of the struts' cross- sectional areas were increased by 20%, and the remaining 1/3 of the 
struts were unchanged. 

The resulting differences in pre-corrupted and post-corrupted model frequencies and 
mode shapes are listed in Table 7.3. The second column represents the percent differences in 
the frequencies of the two models. The third column represents the root mean squared 
(RMS) values of the absolute differences in the mode shapes of the two models. The 
differences between the pre- and post-corrupted model mode shapes are shown pictorially in 
Figure 7.12. The true modes are plotted along the horizontal axis and the corrupted modes 
are plotted along the vertical axis. 



81 



AA = change in cross-sectional area 




Figure 7.11 Model Error Added to MPI FEM 





2 -0.1 0.1 0.2 -0.2 -0.1 0.1 0.2 



Figure 7.12 True vs. Corrupted MPI Mode Shapes 



82 



Table 7.3 Difference Between Pre- and Post-Corrupted Model Frequencies and Mode 

Shapes 



MODE 


Frequency % difference 


Mode Shape RMS values 


1 


3.15 


0.90 e-3 


2 


3.23 


5.20 e-3 


3 


2.01 


5.20 e-3 


4 


3.27 


4.70 e-3 


5 


1.27 


5.60 e-3 


6 


1.27 


10.9 e-3 


7 


1.76 


13.9 e-3 


8 


0.45 


11.8 e-3 


9 


2.23 


9.00 e-3 


10 


3.47 


10.4 e -3 


11 


0.12 


14.7 e-3 


12 


1.96 


14.1 e-3 



7.5.1 Excitation Placement with Model Error 

Once error was introduced into the MPI FEM, the GMIF and GCON excitation 
placement techniques using the corrupted FEM were run. The GMIF derived excitations for 
the uncorrupted and corrupted models are pictured in Figure 7.13, and the GCON derived 
excitations for the uncorrupted and corrupted models are pictured in Figure 7. 14. The GMIF 
excitation placement changed slightly when the model error was added; only node 77 
switched to node 76 when model error was added. The GCON excitation moved the shaker 
from the mid span to the tip of the left boom. The directions for all of the exciters were 
changed when model error was added. 

In order to evaluate the excitations obtained using the corrupted FEM, the time response 
of the MPI structure to impacts at the new excitations was numerically simulated using the 
original uncorrupted model. This time response was then partitioned to the uncorrupted 
unconstrained EI sensor set of section 7.4.1 and was sent to ERA for identification. As in the 
case of the uncorrupted model excitations, the target frequencies and mode shapes were 



83 



sensor 



node 77 

0.5225 X 
0.8323 y 
0.1538 z 



node 19 
0.2471 X 
0.9618 y 
0.1178 z 




node 76 

-0.5890 X 

-0.5714 y 

0.5714 z 



node 19 
-0.5036 X 
0.6800 y 
0.5329 z 



Corrupted FEM 



Figure 7.13 GMIF Derived Excitation Locations 

successfully identified based on percent difference and cross-orthogonality calculations. 
Based on these results, the error added to the FEM had little to no effect on the excitation 
placement configurations' success in exciting the uncorrupted target mode shapes of the 
structure for both the GMIF and GCON excitation placement techniques. 

7.5.2 Sensor Placement with Model Error 

Both the unconstrained and triaxially constrained sensor placement problems were 
evaluated after error was added to the FEM using the six placement techniques previously 
discussed. The changes in sensor set configurations for the unconstrained and constrained 
sets are shown pictorially in Figure 7.15 and Figure 7.16. The original sensors placed using 
the uncorrupted FEM are represented by the boxes. Any sensors that were removed from the 
original sensor set after model error was introduced are represented by circles and any 



84 



node 67 

0.000 X 

0.000 y 

-1.000 z 



node 79 

0.4330 X 

0.2500 y 

-0.8660 z 



node 6 

0.8660 X 

-0.5000 y 

0.0000 z 




Corrupted FEM 



Figure 7. 14 GCON Derived Excitation Locations 

sensors that were added to the original set after model error was introduced are represented 
by triangles. 

The total numbers of sensors that changed for the unconstrained and triaxially 
constrained sensor sets after model error was introduced are listed in Table 7.4. The general 
distribution of the sensors was mostly maintained after model error was added for all of the 
unconstrained sensor sets except for the MIF sensor set. For the constrained sensor sets, five 
of the six placement techniques resulted in a changed sensor set after model error was added. 
The EI technique moved one triax-set from the main boom tip to mid-boom. The KE 
technique moved one triax-set from the left extending boom tip to mid-extending-boom, and 
the EVP technique moved a triax-set from the left extending boom to the main boom. The 
AKE technique resulted in no sensor change after model error was added. The OBS and MIF 



85 




Effective Independence 




Eigenvector Product 




Average Kinetic Energy 



Observability 



MEF 




D original sensor set O removed using model error A added using model error 

Figure 7.15 Model Error Effect on Unconstrained MPI Sensor Sets 



86 



Effective Independence 



Eigenvector Product 




Kinetic Energy 



Average Kinetic Energy 




original triaxial sensors A added using model error 

• removed using model error 
Figure 7.16 Model Error Effect on Constrained MPI Sensor Sets 



87 



technique changed over half the triaxially constrained sensors. For both the unconstrained 
and triaxially constrained cases, the MIF sensor placement technique was particularly 
sensitive to FE model error. 

The original uncorrupted FEM response to the GMIF derived excitation was used to 
evaluate the new sensor sets obtained with the corrupted FEM. The time response discussed 
in the previous section was partitioned to the new sensor configurations and ERA was used to 
identify mode shapes and frequencies. Both the unconstrained and constrained sensor sets 
obtained using the corrupted FEM were successful in identifying the target frequencies 
within 1%, for all six techniques evaluated. The resulting cross-orthogonalities between 
identified (using error sensor sets) and original FEM mode shapes were calculated and are 
pictured in Fig. 7.17 and Fig. 7.18. 

Table 7.4 Number of sensors or triax sets that change when model error is added 



MPI Sensor Set 


EI 


KE 


AKE 


EVP 


OBS 


MIF 


unconstrained 


2 of 18 


2 of 18 


lof 18 


6 of 18 


2 of 18 


16 of 18 


constrained 


lof6 


lof6 


Oof 6 


2 of 6 


3 of 6 


4 of 6 



For the unconstrained sensor sets, the EI, KE, AKE, and MIF techniques resulted in 
generally acceptable cross-orthogonality values for the twelve target modes shown in Fig. 
7.17. Only a few off-diagonal entries of the cross-orthogonalities resulting from these sensor 
configuration were above the acceptable limit of <0.02 for primary modes, but were still 
within the acceptable limit of <0. 10 for secondary modes. The error added to the model in 
this example had little effect on the placement techniques' success in identifying sensor 
configurations which resulted in successful modal information identification. The EVP and 
OBS techniques resulted in poor cross-orthogonalities as was the case when the uncorrupted 
model was used. 

For the triaxially constrained sensor configuration, the model error did not greatly affect 
the uncorrupted cross-orthogonality results for the EI, KE, AKE, and MIF techniques, as 



88 



shown in Fig. 7.18. Even though these cross-orthogonalities are not as good as those 
obtained with no model error, they still lie within acceptable limits previously discussed. 
Therefore, the error added to the model in this example had little effect on the constrained 
placement problem for the EI, KE, AKE, and MIF techniques. For the EVP technique, the 
' model error resulted in a new triaxially constrained configuration which out-performed the 
uncorrupted configuration by chance as can be seen by the cross-orthogonalities pictured in 
Fig. 7.18. This is probably due to the fact that a triax-set was moved to the previously 
uninstrumented main boom. Even though an improvement can be seen here, several of the 
cross-orthogonalities are above the acceptable limits for second mode identification (>0. 10). 
The OBS sensor set cross-orthogonalities showed deterioration after model error was added, 
especially for higher modes. 

7.6 Computational Cost 

The size of the FEM used in both the excitation and sensor placement techniques is the 
basic factor in the computational cost of each technique; the larger the FEM, the greater the 
computational cost of selecting the excitations and sensors. This computational effort is 
generally worthwhile, compared to the cost of planning, implementing, and performing a 
modal test. The approximate floating point calculations for each of the techniques explored 
is given in Table 7.5. 

For the excitation placement using the GMIF, the most expensive part of the calculation 
is the MIF calculation for each possible design. As the GA searches for candidate excitation 
locations, the MIF for each possible location must be calculated in order to evaluate the 
objective function. For larger FEM some steps may be taken to reduce the computation. The 
controllability calculation is not nearly as costly as the MIF calculation. 

Most of the sensor placement techniques are less computationally intensive. For the 
example used in this work, the mass matrix was (240x240) DOFs and the target mode matrix 
was (240x12). The 8-bay truss example used in Chapter 6 had a mass matrix of (80x80) 



89 



DOFs and a target mode matrix of (80x5). The computational costs of the EVP, KE and 
AKE, and EI techniques were on the order of 10^, 10^, and 10^ MATLAB flops. Therefore, 
an increase in computational cost of approximately order 2 can be seen when the number of 
DOFs and the number of target modes are approximately tripled. 

Table 7.5 Floating Point Calculations for MPI Sensor and Excitation Placement 



Placement Technique 


Type of 
Placement 


Floating Point 
Calculations 


G A/Mode Indicator Function 


excitation 


108 


GA/ControUability 


excitation 


106 


Effective Independence 


sensor 


107 


Kinetic Energy 


sensor 


106 


Average Kinetic Energy 


sensor 


106 


Eigenvector Product 


sensor 


103 


Observability 


sensor 


104 


Mode Indicator Function 


sensor 


107 



One way to reduce the computational cost of the techniques evaluated, especially 
effective independence and GMIF techniques, is to reduce the initial set of candidate DOFs 
to a target set. For the size example used in this work, this reduction is not essential, but a 
reduction may be needed for larger models. 



7.7 Conclusions 

A comparative study of several pre-modal-test planning techniques was presented using 
the JPL/MPI testbed. Mode indicator functions calculated using a reduced FEM of the MPI 
structure were used in conjunction with a GA to find location and orientation of two 
excitation sources in order to optimally excite a chosen range of FE target modes during a 
modal test. The original, GMIF, and GCON excitation locations were compared using the 
MPI's simulated structural response to impulses applied at the exciter locations. Both the 



90 



GMIF and GCON excitation locations resulted in a time response to an impulse from which 
the 12 target modes were successfully identified. The original excitation locations resulted 
in a time response to an impulse from which there was a problem extracting modes 5 and 10. 
It should be noted that the original excitation was chosen by an experienced team of 
experimentalists/analysts, using a careful examination of the modes shapes to define a set of 
candidate excitations, along with MIFs to select and verify the final set. Despite this, there is 
still some room for improvement. This indicates the utility of a suite of pre-test planning 
tools to assist the designer, improving the efficiency and completeness of the process. 

Effective independence, kinetic energy, eigenvector product techniques, observability, 
and mode indicator function techniques were used to place a combination of sensors on the 
structure for the purpose of modal identification in two ways: independent sensor placement 
and triaxially constrained placement. For the unconstrained and triaxially constrained 
sensor configurations the EI, KE, and AKE techniques were successful in indentifying the 
target modes and frequencies from the noisy time response. The EI technique resulted in the 
best identification for the unconstrained set and the KE technique resulted in the best 
identification for the constrained set. The OBS and MEF were moderately successful in 
identifying target modes and frequencies from noisy data. The EVP technique was not 
successful in identifying the target modes for either the unconstrained or triaxially 
constrained sets. 

Error was added to the FEM of the MPI structure in order to evaluate its effect on the 
placement techniques. Based on the amount of error added in this example, there was little 
effect seen on the excitation placement techniques, GMIF and GCON. Both techniques 
resulted in an excitation selection using the corrupted model that was successful in exciting 
the target modes. Most of the sensor placement techniques were not greatly affected by the 
introduction of error into the model, with the exception of the MIF sensor placement 
technique. 



91 



a) Effective Independence 




0.10 
0.02 



Figure 7.17 Cross-Orthogonality Between MPI Identified Modes Using Corrupted 
Model and Uncorrupted FEM Modes (18 Unconstrained Sensor Sets) 



92 



d) Eigenvector Product 



n 1.00 
n<0.75 
il]<0.50 

■ <0.25 

■ <0.10 

■ <0.02 



e) Observability 



D 1.00 

■ <0.10 

■ <0.02 




n 1.00 
n<0.75 
El <0.50 

■ <0.25 

■ <0.10 

■ <0.02 



f) Mode Indicator Function 



0.10 
0.02 



modes 



Figure 7.17 — continued 



93 



a) Effective Independence 



1.00 



0.10 
0.02 



D 1.00 
9 <0.10 
■ <0.02 

b) Kinetic Energy 



D 1.00 

■ <0.10 

■ <0.02 




n 1.00 

■ <0.10 

■ <0.02 



c) Average Kinetic Energy 



1.00 



0.10 
0.02 



modes 



Figure 7.18 Cross-Orthogonality Between MPI Identified Modes Using Corrupted 
Model and Uncorrupted FEM Modes (6 Triaxially Constrained Sensors) 



94 



d) Eigenvector Product 



1.00 




n 1.00 
D <0.75 
n<0.50 
n <0.25 

■ <0.10 modes 

■ <0.02 

f) Mode Indicator Function 



D 1.00 
E <0.75 

ia<o.5o 

■ <0.25 

H<o.io 

■ <0.02 



0.10 
0.02 



modes 



Figure 7.18 — continued 



CHAPTER 8 
PRE-MODAL TEST PLANNING ALGORITHM APPLICATION: 

CAR BODY 



8.1 Introduction 

In an effort to further examine the sensor and actuator placement strategies developed in 
Chapters 4 and 5 a car body FEM is considered. Recently, Came and Dohrmann used this car 
FEM to explore modal test design strategies for the purpose of FE model correlation (1995). 
The excitation and sensor placement techniques developed in Chapter 5 are used on the same 
car FEM and the results are compared to those obtained by Carne and Dohrmann in their 
work. 

8.2 Excitation Placement 

Three candidate inputs were chosen by the modal test engineers based on intuition and a 
knowledge of the mode shapes of the car. Mode indicator functions were calculated for each 
of the shakers to insure their success in exciting the first 10 non-rigid body modes of the 
system. The GMIF and GCON algorithms were also used to place three excitation devices on 
the car body. The location of the shakers on the car body are shown in Figure 8.1 and are 
given in Table 8.1. These forces were constrained to be in any 0, 30, 45, 60, or 90 degree 
orientation in the x,y, and z planes. A comparison of the minimum MIF values of the 3 forces 
for all of the techniques is given in Table 8.2. 



95 



96 



Original shaker location 




GMIF shaker location 



GCON shaker location 




Figure 8. 1 Car Body Shaker Locations 



Table 8.1 Car Body Excitation Location and Orientation 



Excitation 


node 


x-direction 


y-direction 


z-direction 


original 


100331 








1.0 


100309 





1.0 





100778 








1.0 


GMIF 


100129 


-0.5 


0.9 





100226 








1.0 


100780 





0.5 


0.9 


GCON 


100773 


0.2 


-0.4 


0.9 


100598 








-1.0 


100219 








-1.0 



97 



Table 8.2 Mode Indicator Function Values for Various Excitation Placements 



Mode Number 


Modal Test 


GMIF 


GCON 


1 


0.01 


0.01 


0.00 


2 


0.01 


0.00 


0.01 


3 


0.03 


0.07 


0.03 


4 


0.23 


0.12 


0.41 


5 


0.10 


0.12 


0.12 


6 


0.02 


0.10 


0.31 


7 


0.19 


0.09 


0.11 


8 


0.07 


0.04 


0.09 


9 


0.32 


0.14 


0.12 


10 


0.09 


0.04 


0.18 


min MIF 


2/10 


6/10 


2/10 



The GMIF excitation was the only excitation with all MIF values less than 0. 15. The 
original excitation resulted in a maximum MIF value of 0.3 for one of the 10 target modes 
and the GCON excitation resulted in off diagonals of 0.4 and 0.3 for 2 of the 10 target modes. 
Five hundred three-shaker excitation sets were generated randomly in order to compare the 
three excitation placements to other excitation possibilities on the structure. Five hundred 
random shakers were chosen because approximately 500 population members were 
evaluated when the GMIF algorithm was run. A MIF for the 10 target modes for each of the 
three forces was taken and the minimum MIF for each mode was retained for the 500 
randomly located excitation sets. 

The MIF values for the randomly placed shakers as well as those being compared are 
pictured in Figure 8.2. In Figure 8.2 the MIF values of the 500 random excitation sources are 
sorted and graphed. The values of the three techniques being compared are superimposed on 
top of each graph. The top portion of Figure 8.2 is a graph of the minimum MIF values, the 
middle portion of the figure is a graph of the maximum MIF values, and the bottom portion is 
a graph of the sum of the MIF values for the 10 target modes (i.e., GMIF algorithm objective 



98 



function value). The key portion of Fig. 8.2 is the center portion, maximum MIF value, since 
it is desired that the maximum MIF of the union of all excitation forces in an excitation set be 
as small as possible. The maximum MIF of the GMEF excitation placement is lower than 
1 00% of the randomly placed sensor sets . The highest maximum MIF of the three excitations 
is the GCON excitation placement, which is higher than approximately 60% of the randomly 
placed excitation sets. 

The angle between the sub-spaces of the mode shapes and the supspaces of the input 
direction cosines was used in the GCON algorithm to find an excitation location which 
minimizes this angle for all of the target modes. The degree of controllability angle was 
calculated for the 500 randomly generated excitations as well as for original and GMIF 
excitation locations. Graphs of these values are pictured in Figure 8.3. The top portion of 
Figure 8.3 is the minimum degree of controllability angle, the middle portion is a graph of the 
maximum degree of controllability angle, and the bottom graph is a sum of the degree of 
controllability angles for the target modes (i.e., GCON algorithm objective function value). 
The GCON excitation placement outperformed all of the random excitations based on the 
controllability calculations. A comparison of the degree of controllability angles for each 
excitation is given in Table 8.3. 

Based on the comparison of the MIF values, the three techniques used to place excitation 
devices performed better than a majority of randomly placed excitation set. The genetic 
algorithm used to drive the MIF and controllability excitation placement algorithms resulted 
in excitation locations which outperformed a random set of excitations based on the values of 
the designed objective functions. The question arises will the techniques be computationally 
feasible for larger FEM. Table 8.4 gives the floating point calculation required to place the 
excitation sets using the car body FEM. The GMIF technique resulted in a more successful 
excitation set based on MIF values than an exhaustive search of 500 random excitation sets, 
and both techniques required the same floating point calculations. 



99 



0.03 



0.02- 






0.01 



Minimum MEF Value 




150 200 250 300 

Random Members 



random 



original 

GCON 

GMIF 



500 



4 


Maximum MIF Value 


0.8 




, 0.6 

So.4 


^^^-^^'^^ 


___^ 


0.2 


>^— ' 


r\ 


1 1 1 1 1 1 1 1 1 



50 100 150 200 250 300 

Random Members 



350 



400 



450 



random 



GCON 

original 

GMIF 



500 



2.5- 

^1.5 



0.5 



Sum of MIF of target modes 






50 100 150 200 250 300 

Random Members 



350 



400 



450 



GCON 

- original 



GMIF 



500 



Figure 8.2 MIF Values for Excitation Placements Compared to 500 
Randomly Located 3-Point Excitations 



100 




Minimum Controllability Angle 



n 1 1 r- 



-| r 



_J L_ 



random 
GMIF 

original 
GCON 



50 100 150 200 250 300 350 400 450 500 

Random Members 



Maximum Controllability Angle 



random 




GCON 



50 100 150 200 250 300 350 400 450 500 ' 

Random Members 

, ■ ■ ■■■■■■> f^ - 



1000 
o 980 

t-H 

/S^960 



940 



920 



900 



Genetic Controllability Objective Function Value 






random 



GMIF 



original 



50 100 150 200 250 300 350 400 450 500 

Random Members 



GCON 



Figure 8.3 Controllability Values for Excitation Placements Compared to 
500 Randomly Located 3-Point Excitations 



101 



Table 8.3 Controllability Angles (in degrees) for Excitation Placements 



Mode Number 


Modal Test 


GMIF 


GCON 


1 


86.3 


88.7 


86.6 


2 


88.5 


87.3 


86.6 


3 


79.9 


89.2 


84.3 


4 


88.3 


88.7 


82.0 


5 


84.3 


87.6 


78.3 


6 


88.3 


88.6 


78.9 


7 


85.8 


87.2 


73.4 


8 


84.8 


87.1 


72.7 


9 


84.9 


87.2 


86.3 


10 


83.4 


86.2 


87.3 


min controllability 
angle 


2/10 


0/10 


8/10 



Table 8.4 Excitation Placement Techniques Floating Point Calculations 



Excitation Placement Technique 


Floating Point Calculations 


GMIF 


109 


GCON 


107 


500 Random 


109 



In order to reduce the computational cost of the GMIF technique, a method of reducing 
the FEM DOFs down to a candidate set should be explored. The maximum kinetic energy or 
maximum average kinetic energy would be a good candidate. The full FEM mass and mode 
shape matrices would be used to calculate the KE energy matrix at a cost of m(2n2+n) 
floating point calculations (where n is the number of DOFs and m is the number of target 
modes). The DOFs with the maximum kinetic energy or maximum average kinetic energy 
over all of the target modes will be chosen as the reduced candidate DOFs for excitation 



102 



placement. The FE mode shape matrix will be partitioned to the reduced candidate DOF set 
and sent to the GMIF excitation placement algorithm. 

8.3 Sensor Placement 

Sensors are placed on the car FEM using effective independence, observability, and MIF 
techniques. The sensors being placed are 34 triaxially constrained sensor sets in order to 
compare the placements to the original intuition set placed on the car body. The three derived 
sensor sets and the intuition sensor set are shown in Figure 8.4. 



Intuition Sensor Set 



Effective Independence 




Observability 



Figure 8.4 34 Triaxially Located Car Body Sensor Sets 



103 



The triaxially constrained sensor sets placed with the effective independence and 
observability techniques are very similar. The observability technique collocated 24 of the 
34 triax sensors with the effective independence set. A majority of the sensors were placed 
on the roof, the package shelf, and on the front and rear ends. The intuition set distributed the 
sensors over the entire body. The effective independence technique is much more 
computationally intensive than the observability technique, which is due to the calculation of 
the eigensolution of the Fisher information matrix over each iteration. Table 8.5 gives a 
comparison of the floating point calculation required to place 34 triaxially located sensors on 
the car FEM. The observability technique is less computationally intensive. Based on the 
similarities of the EI and observability sensor configurations, the observability sensor 
placement technique may be used as a tool to establish a reduced set of candidate sensor 
DOFs or nodes for the effective independence technique to choose from. Both of the 
techniques tend to cluster sensors in specific locations. This is due to the fact that more 
sensors than target modes are being placed the structure. Once the EI and observability 
techniques find DOFs which indicate good target mode linear independence or observabihty, 
they tend to continue to place additional sensors in that region even though redundant 
information is being obtained. It is recommended that the approximate number of sensors to 
be placed by these techniques is equal to or slightly greater than the number of target modes 
to be identified. Any additional sensors required should be placed with another technique(s) 
such as the MIF technique. 

The MIF sensor placement algorithm is the most computationally intensive of the three 
sensor placement techniques due to the iterative MIF calculations of the various sensor sets. 
However the distribution of the sensors placed using MIF calculations is more even over the 
entire body. This allows for better mode shape visualization, which is an important aspect of 
modal test data. The MIF technique did not place any sensors on the roof package of the car 
which contradicts the EI and observability sets. The MIF sensor configuration would 



104 



therefore be a good compliment to the EI sensor sets due to the new information it would 
provide about the car body. 

Table 8.5 Triaxial Sensor Placement Techniques Floating Point Calculations 



Sensor Placement Technique 


Floating Point Calculations 


MIF 


108 


Effective Independence 


108 


Observability 


105 



■l^' 



CHAPTER 9 
CONCLUSIONS AND FUTURE WORK 



This study investigated the use of genetic algorithms in the areas of finite element model 
refinement and pre-modal test planning excitation and sensor placement. A genetic 
algorithm based model refinement technique was developed. The purpose of the algorithm 
was to perturb FEM parameters to obtain an updated FEM with measured modal properties. 
The algorithm was examined using a simple FEM of a truss. The FEM of the truss was 
updated using numerically simulated modal data. The GMRA resulted in an updated FEM 
with modal properties very close to the measured modal properties. For the size of the 
example performed the GMRA was very effective and cost efficient. However, for very 
large FEMs, GMRAs true utility may lie in mapping out the design space. GMRA may be 
used to identify which design variables need to be changed or to locate general areas in the 
design space for gradient search focus. 

The second scope of this study was to develop new excitation and sensor placement 
techniques for vibration test planning, some of which utilize genetic algorithms. An 
excitation technique which used the FEM to calculate a normal MIF was developed. A 
genetic algorithm was used to search a design space for excitation locations and orientations 
which exhibit sharp MIF drops for target modes. A sharp drop in MIF values at the target 
modes indicates that those modes are well excited. The GMIF excitation placement 
algorithm was evaluated on 2 FEM of increasing complexity and proved to be a powerful 
excitation placement tool. For the examples explored, it was both robust to noise and 
outperformed large random excitation selections. One drawback of GMIF is that it will be 
computationally intensive for large models. Some of the techniques explored such as kinetic 
energy may be used to reduce the full FEM down to a candidate set of node points for the 



105 



106 



GMIF algorithm to choose from. A sensor placement algorithm was also developed using 
the FE MIF and was evaluated using 2 FEMs. The MIF sensor sets were the most unusual 
sensor configurations compared to the other placements examined in this study. Regardless 
of the unusual placements, the MIF sensor sets were successful in identifying the target 
modes required. The MIF sensor techniques distributed sensors over more of the structures 
than the other techniques which tended to cluster sensors in particular areas. This even 
distribution is well suited for mode shape visualization and possibly for damage detection 
purposes. However, the MIF sensor sets were particularly sensitive to FEM error. Even 
though the sensor sets changed after the model was corrupted, they still resulted in successful 
system identification for the examples examined. This result is possibly due to the overall 
distribution of the MIF sensor sets. 

The concepts of modal controllability and observability were used to develop additional 
excitation and sensor placement algorithms. The degree of re mode shape controllability 
was used with a genetic algorithm to search for excitation locations and orientations with 
high modal controllability. An excitation location with a high controllability for all of the 
target modes will be the best location to put energy into a system for a vibration test. The 
GCON algorithm was evaluated using 3 FEMs. For the examples explored, the GCON 
excitations outperformed a large random sample of excitations. The degree of FE mode 
shape observability was used to develop an optimal sensor location algorithm. DOFs with 
high target mode shape observability are chosen as sensor locations because the mode shapes 
are most observable at those points. The observability sensor placement algorithm was 
evaluated using 3 FEM. In general the observability technique did not locate sensor sets 
which were successful in identifying all of the target modes of the system. However, in the 
examples explored, the observability technique collocated a large number of sensor with the 
more computationally intensive effective independence technique. The observability 
technique may be used to reduce a large number of FE DOFs down to a candidate set for the 
EI technique to choose from. 



107 



Overall, the genetic algorithm was shown to be a feasible tool for the purpose of 
excitation placement. Both the GCON and GMIF algorithms resulted in excitation 
selections which outperformed random sets of excitations which required more calculations 
and intuition excitation placements. The most efficient use of these tools would be to 
compliment an intuition set of shakers and sensors being used in a modal test. For example, 
the GMIF algorithm may be used to search for an additional shakers to make up for any high 
target mode MIF values which are observed in an intuition set as was the case for the MPI 
truss. Similarly, many of the sensor placement techniques may be used together to 
compliment an intuition set developed by a test designer. For example m sensors may be 
placed using effective independence to assure hnear independence of the m target modes to 
be measured, and the remaining sensors may be placed using the MIF technique to assure a 
more even distribution of the sensors. Regardless of the techniques used, all of the sensor and 
excitation placement algorithms are entirely dependent on the FEM of the structure that the 
modal test is being designed for. Therefore, the tools are only as good as the model they are 
based on and no more confidence than the designer has in the model should be afforded to the 
tools which utilize it. The true utility in these tools lies in their ability to expand on those 
sensor and excitation selections which are based on the intuition of the modal test designer. 

Future work for this study would be an evaluation of all of excitation and sensor 
placement techniques using a true vibration test. All of the techniques would be used to place 
sensor and excitation devices on a structure. Various modal test should be performed to 
obtain the appropriate data for each of the techniques. The excitation placements may be 
evaluated on how well the target modes were excited relative to one another, and the sensor 
placement techniques may be evaluated on how well they extracted each of the target modes. 
The modal data may then be used to update the FEM of the structure. Each of the excitation 
and sensor placement techniques may be rerun using the updated FEM. The final evaluation 
would be the sensitivity of each of the techniques to the change in the FEM. 



REFERENCES 



Adelman, H. M. and Haftka, R. T. (1986), "Sensitivity Analysis of Discrete Structural 
Systems," AIAA Journal, Vol. 24, No. 5. pp. 823-832. 

Andry, A. N., Shapiro, E. Y., and Chung, J. C. (1983), "Eigenstructure Assignment For 
Linear Systems," IEEE Transactions on Aerospace And Electronic Systems, Vol. 
AES-19,No. 5, pp. 711-729. 

Baruch, M., and Bar Itzhack, I.Y. (1978), "Optimum Weighted Orthogonalization of 
Measured Modes," AIAA Journal, Vol. 16, No. 4, pp. 346-351. 

Berman, A. and Flannelly, W. G., (1971), "Theory of Incomplete Models of Dynamic 
Structures," AIAA Journal, Vol. 9, No. 8, pp. 1481-1487. 

Berman, A. and Nagy, E. J. (1983), "Improvements of a Large Analytical Model Using Test 
Data," AMA Journal, Vol. 21, No. 8, pp. 1168-1173. 

Came, T. G., Mayes, R. L., and Levine-West, M. B. (1993), "A Modal Test of a Space-Truss 
for Structural Parameter Identification," Proceedings of the 11'^ International Modal 
Analysis Conference, Kissimee, FL, pp. 486-494. 

Chen, J. C. and Garba, J. A. (1980), "Analytical Model Improvement Using Modal Test 
Results," AMA Journal, Vol. 26, No. 12, pp. 1119-1126. 

Collins, J. D., Hart, G. C, Hasselman, T. K. and Kennedy, B. (1974), "Statistical 
Identification of Structures," AIAA Journal, Vol. 12, No. 2, pp. 185-190. 

Creamer, N. G., and Hendricks, S. L. (1987), "Structural Parameter Identification Using 
Modal Response Data," Proceedings of the 6* VPI&SU/AIAA Symposium on 
Dynamics and Controls for Large Structures, Blacksburg, VA, pp. 27-38. 

Ewins, D. J. (1984), Modal Testing: Theory and Practice, John Wiley & Sons, New York. 

Flanigan, C. C. (1991), "Correction of Finite Element Models Using Mode Shape Design 
Sensitivity," Proceedings of the 9^** International Modal Analysis Conference, Firenza, 
Italy, pp. 151-159. 



108 



109 



Flanigan, Christopher C, and David L. Hunt (1993), "Integration of Pretest Analysis and 
Model Correlation Methods for Modal Surveys," Proceedings of the 1 1^'^ International 
Modal Analysis Conference, Kissimee, FL, February, pp 444-448. 

Fuh J., Chen S. and Berman A. (1984), "System Identification of Analytical Models of 
Damped Structures," Proceedings of the 25^'^ AIAA Structures, Structural Dynamics 
and Materials Conference, Palm Springs, CA, pp. 112-122. 

Goodson, R. E., and Polls, M. P. (1978), "Identification of Parameters in Distributed 
Systems," Distributed Parameter Systems, edited by W. H. Ray and D.G. Lainiotis, 
Dekker, New York. 

Grefenstette, J. J. (ed.), (1987), Proceedings of the Second International Conference on 
Genetic Algorithms and Their Applications, Lawrence Erlbaum Associates, Hillsdale, 
NJ. 

Hamdan, A. M. A., and Nayfeh, A. H. (1989), "Measures of Modal Controllability and 
Observability for First- and Second-Order Linear System," Journal of Guidance and 
Control, Vol. 12, No. 3, pp. 421-^27. 

Hanagud, S., Meyyappa, M. Cheng, Y. P, and Graig, J. I. (1984), "Identification of 
Structural Dynamics Systems with Nonproportional Damping," Proceedings of the 25^** 
AIAA Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, 
pp. 283-291. 

Heylen, W., and Sas, R (1987), "Review of Model Optimization Techniques," Proceedings 
of the 5th International Modal Analysis Conference, Orlando, FL, pp. 1 177-1 182. 

Holland, J. H. (1975), Adaptation in Natural and Artificial Systems, The University of 
Michigan Press, Ann Arbor. 

Hunt, D. L., Void, H., Peterson, E. L., and Williams, R. (1984), "Optimal Selection of 
Excitation Methods for Enhanced Modal Testing," Proceedings of the 25^'^ AIAA 
Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, pp. 
1023-1030. 

Ibrahim, S.R. and Saafan, A.A., (1987), "Correlation of Analysis and Test in Modeling of • 
Structures, Assessment and Review," Proceedings of the 5th International Modal 
Analysis Conference, Orlando, FL, pp. 1651-1660. 

Inman, D.J. and Minas, C. (1990), "Matching Analytical Models with Experimental Modal 
Data in Mechanical Systems," Control and Dynamics Systems, Vol. 37, pp. 327-363. 

Jarvis, B. (1991), "Enhancement to Modal Testing Using Finite Elements," Sound and 
Vibration, Vol. 8, pp. 28-30. 

Juang, J., and Pappa, R. (1985), "An Eigensystem Realization Algorithm for Modal 
Parameter Identification and Model Reduction," Journal of Guidance and Control, Vol. 
8, pp. 620-627. 



no 



Juang, J., and Rodriguez, G. (1979), "Formulations and Applications of Large Structure 
Actuator and Sensor Placements," Proceedings of the 2"^^ VPI&SU/AIAA Symposium 
on Dynamics and Control of Large Flexible Spacecraft, VPL Blacksburg, VA, pp. 
247-262. 

Kabe, A. M. (1985), "Stiffness Matrix Adjustment Using Mode Data," AlAA Journal, Vol. 
23, No. 9, pp. 1431-1436. 

Kailath, T. (1980), Linear Systems, Prentice-Hall, Englewood Cliffs, NJ. 

Kammer, D. C. (1991), "Sensor Placement for On-Orbit Modal Identification and 
Correlation of Large Space Structures," Journal of Guidance, Control, and Dynamics, 
Vol. 15,No. 2, pp. 251-259. 

Kammer, D. C. (1987), "Test-Analysis Model Development Using an Exact Model 
Reduction," International Journal ofAnalytical and Experimental Modal Analysis,'Vol. 
2, No. 4, pp. 175-179. 

Kaouk, M. (1993), "Structural Damage Assessment and Finite Element Model Refinement 
Using Measured Modal Data," dissertation. University of Florida, Gainesville. 

Kashangaki, T. A. L. (1992), "Ground Vibration Tests of a High Fidelity Truss for 
Verification of on Orbit Damage Location Techniques," NASA Technical 
Memorandum 107626, May. 

Kientzy, D., M. Richardson, and K. Blakely (1989), "Using Finite Element Data to Set Up 
Modal Tests," Sound and Vibration, Vol. 6, pp 16-23. 

Larson, C. B., Zimmerman, D. C, and Marek, E. L. (1994), "A Comparison of Modal Test 
Planning Techniques: Excitation and Sensor Placement Using the NASA 8-Bay Truss," 
Proceedings of the 12^^ internadonal Modal Analysis Conference, Honolulu, HI, pp. 
205-211. 

Laub, A. J. and Arnold, W. F. (1984), "Controllability and Observability of Linear Matrix 
Second-Order Models," IEEE Transactions on Automatic Control, Vol. AC-29, No. 2, 
pp. 163-165. 

Le Pourhiet, A., and Le Letty, L. (1978), " Optimization of Sensor Locafions in Distributed 
Parameter System Identification," Identification and System Parameter Estimation, 
North Holland Publishing, Amsterdam, pp. 1681-1592. 

Levine-West, M., Kissil, A., and Milman, M. ( 1 994), "Evaluation of Mode Shape Expansion 
Techniques on the Micro-Precision Interferometer Truss," Proceedings of the 12'*' 
International Modal Analysis Conference, Honolulu, HA, pp. 212-218. 

Martinez, D., Red-Horse, J. and Allen, J. (1991), "System Idendficafion Methods for 
Dynamic Structural Models of Electronic Packages," Proceedings of the 32"'^ AIAA 



Ill 



Structures, Structural Dynamics and Materials Conference, Baltimore, MD, pp. 
2336-2346. 

Matzen, V.C. (1987), "Time Domain Identification of Reduced Parameter Models," 
Proceedings of the SEM Spring Conference on Experimental Mechanics, Houston, 
Texas, pp. 401^08. 

O'Callahan, J.C, Avitabile PA. and Riemer, R. (1989), "System Equivalent Reduction 
Expansion Process (SEREP)," Proceedings of the 7''' International Modal Analysis 
Conference, Las Vegas, NV, pp. 29-37. 

Omatu, S., Koide, S., andSoeda,T. (1978), "Optimal Sensor Location for Linear Distributed 
Parameter Systems," IEEE Transactions on Automatic Control, Vol. AC-23, No. 4, pp. 
665-673. 

Qureshi, Z. H., and Goodwin, G. C. (1980), "Optimum Experimental Design for 
Identification of Distributed Parameter Systems," International Journal of Control, 
Vol. 31, No. 1, pp. 21-29. 

Red-Horse, J. R., Marek, E. L., Levine-West, M. (1993), "System Identification of the JPL 
Micro-Precision Interferometer Truss: Test-Analysis Reconciliation," Proceedings of 
the 34'^ AIAA/ASME Structures, Structural Dynamics, and Materials Conference, 
LaJolla, CA, pp. 3353-3365. 

Rodden, W. P. (1967), "A Method for Deriving Structural Influence Coefficients from 
Ground Vibration Tests," AIAA Journal, Vol. 5, No. 5, pp. 991-1000. 

Salama, M., Rose, T., and Garba, J. (1987), "Optimal Placement of Excitations and Sensors 
for Verification of Large Dynamical Systems," Proceedings of the 28^*^ AIAA/ASME 
Structures, Structural Dynamics, and Materials Conference, San Diego, CA, pp. 
1024-31. 

Sawaragi, Y., Soeda, T., and Samatu, S. (1978), Modeling, Estimation and Their Application 
for Distributed Parameter Systems, Lecture Notes in Control & Information Science, 
Vol. 11, Springer- Verlag, Berlin, Germany. 

Schaffer, J.D. (ed.) (1989), Proceedings of the Third International Conference on Genetic 
Algorithms and Their Applications, Morgan Kaufmann Publishers, Inc., San Mateo, 
CA. 

Shah, R C. and Udwadia, E E. (1978), " Methodology for Optimal Sensor Locations for 
Identification of Dynamic Systems," Journal of Applied Mechanics, Vol. 45, pp. 
188-196. 

Smith, S. W. and Beattie, C. A., (1990), "Simultaneous Expansion and Orthogonalization 
of Measured Modes for Structure Identification," Proceedings of the AIAA Dynamic 
Specialist Conference, Long Beach, CA, pp. 261-270. 



112 



Udwadia, F. E., and Garba, J. A. (1985), "Optimal Sensor Locations for Structural 
Identification," JPL Proceedings of the Workshop on Identification and Control of 
Flexible Space Structures, April, pp. 247-261. 

Williams, R., Crowley, J., and Void, H., "The Multivarient Mode Indicator Function in 
Modal Analysis," Proceedings of the 3'"'^ International Modal Analysis Conference, 
1985. 

Yu, T. K., and Seinfeld, J. H. (1973), "Observability and Optimal Measurement Locations 
In Linear Distributed Parameter Systems," International Journal of Control, Vol. 18, 
No. 4, pp. 785-799. 

Zimmerman, D. C. (1993), "A Darwinian Approach to the Actuator Placement Problem 
With Nonnegligible Actuator Mass," Mechanical Systems and Signal Processing, Vol. 
7, No. 4, pp. 363-374. 

Zimmerman, D. C. and Smith S. W, (1992), "Model Refinement and Damage Location for 
Intelligent Structures," book chapter in Intelligent Structural Systems, edited by H. S. 
Tzou, Kluwer Academic Publishers, Boston. 

Zimmerman, D.C., and Widengren, W. (1990), "Model Correction Using a Symmetric 
Eigenstructure Assignment Technique," AIAA Journal, Vol. 28, No. 9, pp. 1670-1676. 



BIOGRAPHICAL SKETCH 

Cinnamon Buckels Larson received her Bachelor of Science in Engineering degree 
from the University of Florida department of Aerospace Engineering, Mechanics, and 
Engineering Science in May of 1 990. She received a Master of Science degree from the same 
department in December of 1992. 



113 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



vX.-//^ 




David C. ZimmermaA, Chair 
Associate Professor of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



P^niel Drucker 
'''^Graduate Research Professor Emeritus of 
Aerospace Engineering, Mechanics, and 
Engineering Science 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 




Norman Fitz-Coy 
Assistant Professor of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 




:^/^--^ 



Peter Ifji 

Assistant Professor of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in §iCope and quality, as a 
dissertation for the degree of Doctor of Philosophy^ 




Marc Hoit 

Associate Professor of Civil Engineering 



This dissertation was submitted to the Graduate Faculty of the College of 
Engineering and to the Graduate School and was accepted as partial fulfillment of the 
requirements for the degree of Doctor of Philosophy. 



May 1996 



/' 




^infred M. Phillips 
Dean, College of Engineering 



Karen A. Holbrook 
Dean, Graduate School 



LD 

1780 

1996 



UNIVERSITY OF FL ORina 

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3 1262 08555 0670